GIFT   OF 

Daun:hter   of 


LKGlNE^Rlit 


AN 


ELEMENTARY  TREATISE 


INTEGRAL  CALCULUS 


FOUNDED   ON  THE 


METHOD  OF  RATES  OR  FLUXIONS 


WILLIAM   WOOLSEY   JOHNSON 

PROFESSOR   OF  MATHEMATICS   AT  THE   UNITED   STATES  NAVAL  ACADEMY 
ANNAPOLIS   MARYLAND 


NEW   YORK: 

JOHN    WILEY    AND    SONS, 

53  East  Tenth  Street, 

1892. 


COPYPIGMT, 

1881, 

By  JOHN  WILEY  AND  SONS. 


\\\c^     c\     '-^  ^^"\'-' 


01  FT  OF 

It. 

ENGINEERFNG  LIBRARY 


p«ess  or  i.  i.  LiTTie  &  co., 
M08-  10  TO  aa  ASTon  place,  new  roRI* 


S-i 


PREFACE. 


This  work,  as  at  present  issued,  is  designed  as  a  shorter 
course  in  the  Integral  Calculus,  to  accompany  the  abridged 
edition  of  the  treatise  on  the  Differential  Calculus,  by  Pro- 
fessor J.  Minot  Rice  and  the  writer.  It  is  intended  hereafter 
to  publish  a  volume  commensurate  with  the  full  edition  of  the 
work  above  mentioned,  of  which  the  present  shall  form  a  part, 
but  which  shall  contain  a  fuller  treatment  of  many  of  the  sub- 
jects here  treated,  including  Definite  Integrals,  and  the  Me- 
chanical Applications  of  the  Calculus,  as  well  as  Elliptic  Inte- 
grals, Differential  Equations,  and  the  subjects  of  Probabilities 
and  Averages.  The  conception  of  Rates  has  been  employed 
as  the  foundation  of  the  definitions,  and  of  the  whole  subject 
of  the  integration  of  known  functions.  The  connection  be- 
tween integration,  as  thus  defined,  and  the  process  of  summa- 
tion, is  established  in  Section  VII.  Both  of  these  views  of  an 
integral — namely,  as  a  quantity  generated  at  a  given  rate,  and 
as  the  limit  of  a  sum — have  been  freely  used  in  expressing 
geometrical  and  physical  quantities  in  the  integral  iorw. 


508233 


1 V  PREFA  CE. 

The  treatises  of  Bertrand,  Frenct,  Gregory,  Todhunter,  and 

Williamson,  have  been  freely  consulted.     My  thanks  are  due 

to  Professor  Rice  for  very  many  valuable  suggestions  in  the 

course  of  the  work,  and  for  performing  much  the  larger  share 

^f  the  work  of  revising  the  proof-sheets. 

W.  W.  J. 

U.  S.  Naval  Academy,  July,  1881. 


CONTENTS. 


CHAPTER   I. 

Elementary  Methods  of  Integration. 
I. 

PAGE 
I 

The  differential  of  a  curvilinear  area 3 

Definite  and  indefinite  integrals 4 

Elementary  theorems 6 

Fundamental  integrals 7 

Examples  I lo 


Integrals . 


II. 

Direct  integration 14 

Rational  fractions 15 

Denominators  of  the  second  degree 16 

Denominators  of  degrees  higher  than  the  second 19 

Denominators  containing  equal  roots. . . « 22 

Examples  II 26 

III. 

Trigonometric  integrals 33 

Cases  in  which    sin"'  0  cos"  0  d9  is  directly  integrable 34 


The  integrals 
The  integrals 


,  and 


36 


37 


VI  COiVTEN-TS. 

PAGE 

Miscellaneous  trigonometric  integrals 38 

The  integration  of ; aq 

"  a  +  6  cose  ^ 

Examples  III 43 


CHAPTER  II. 

Methods  of  Integration — Continued, 

IV. 

Integration  by  change  of  independent  variable 50 

Transformation  of  trigonometric  forms 51 

Limits  of  a  transformed  integral 53 

The  reciprocal  of  x  employed  as  the  new  independent  variable 53 

A  power  of  x  employed  as  the  new  independent  variable 54 

Examples  IV 56 


Integrals  containing  radicals 59 

Radicals  of  the  fonn  /t/{ax'^  -V  b) - 61 

The  integration  of          ^ — - 64 

V(-^   ±0.') 

Transformation  to  trigonometric  forms 65 

Radicals  of  the  form  /^[ax^  -ir  bx  +  c) 67 

The  integrals  |  — ^ and     | ^-^ 68 

Examples  V 70 

VI. 

Integration  by  parts 77 

A  geometrical  illustration 78 

Applications 78 

Formulas  of  reduction   Si 


Reduction  of     sin'"  9  ds    and      cos'"  Qd6 82 

Reduction  of     sin'"  flcos"  e</d 84 


CONTENTS.  Vll 


PAGE 

Illustrative  examples _. 87 

Extension  of  the  formula  employed  in  integration  by  parts 8g 

Taylor's  theorem 9^ 

Examples  VI 9^ 

VII. 

Definite  integrals 97 

Multiple- valued  integrals 100 

Formulas  of  reduction  for  definite  integrals loi 

Elementary  theorems  relating  to  definite  integrals 104 

Change  of  independent  variable  in  a  definite  integral 105 

The  differentiation  of  an  integral 106 

Integration  under  the  integral  sign 109 

The  definite  integral  regarded  as  the  limiting  value  of  a  sum iii 

Additional  formulas  of  integration 115 

Examples  VII H7 


CHAPTER    III. 

Geometrical  Applications. 

VIII. 

Areas  generated  by  variable  lines  having  fixed  directions 123 

Application  to  the  witch 124 

Application  to  the  parabola  when  referred  to  oblique  coordinates 126 

The  employment  of  an  auxiliary  variable 126 

Areas  generated  by  rotating  variable  lines 128 

The  area  of  the  lemniscata 129 

The  area  of  the  cissoid 130 

A  transformation  of  the  polar  formulas 130 

Application  to  the  folium 131 

Examples  VIII I34 

IX. 

The  volumes  of  solids  of  revolution 141 

The  volume  of  an  ellipsoid 143*!! 

Solids  of  revolution  regarded  as  generated  by  cylindrical  surfaces 144 

Double  integration 145 

Determination  of  the  volume  of  a  solid  by  double  integration 149 


VIU  COXTENTS. 


PAGB 

The  determination  of  volumes  by  triple  integration 150 

Elements  of  area  and  volume 152 

Polar  elements i  c  < 

The  determination  of  volumes  by  polar  formulas 155 

Polar  coordinates  in  space 157 

Application  to  the  volume  generated  by  the  revolution  of  a  cardioid 159 

ExampUs  IX 160 


X. 

Rectification  of  plane  curves ....  168 

Rectification  of  the  semi-cubical  parabola 16S 

Rectification  of  the  four-cusped  hypocycloid i6g 

Change  of  sign  oi  ds 170 

Polar  coordinates 170 

Rectification  of  curves  of  double  curvature 171 

Rectification  of  the  loxodromic  curve 172 

Examples  X 173 

XI. 

Surfaces  of  solids  of  revolution 178 

Quadrature  of  surfaces  in  general 179 

The  expression  in  partial  derivatives  for  sec  v 180 

The  determination  of  surfaces  by  polar  formulas iSi 

Examples  XI 183 

XII. 

Areas  generated  by  straight  lines  moving  in  planes 186 

Applications 187 

Sign  of  the  generated  area 189 

Areas  generated  by  lines  whose  extremities  describe  closed  circuits 190 

Amsler's  Planimeter 191 

Examples  XII 193 


XIII. 

Approximate  expressions  for  areas  and  volumes 195 

Simpson's  rules 197 

Cotes'  method  of  approximation 198 


CONTENTS.  IX 

PAGE 

Weddle's  rule igg 

The  five-eight  rule igg 

The  comparative  accuracy  of  Simpson's  first  and  second  rules 200 

The  application  of  these  rules  to  solids 200 

Woolley's  rule 201 

Examples  XIII 202 


CHAPTER    IV. 

Mechanical  Applications. 

XIV. 

Definitions 204 

Statical  moment 204 

Centres  of  gravity 206 

Polar  formulas 208 

Centre  of  gravity  of  the  lemniscata 2og 

Solids  of  revolution 209 

Centre  of  gravity  of  a  spherical  cap 2icr- 

The  properties  of  Pappus 210 

Examples  XIV 212 

XV. 

Moments  of  inertia 21Q 

Moment  of  inertia  of  a  straight  line 220 

Radii  of  gyration 220 

Radius  of  gyration  of  a  sphere 221 

Radii  of  gyration  about  parallel  axes 222 

Application  to  the  cone 223 

Polar  moments  of  inertia 225 

Examples  XV 225 


THE 

INTEGRAL^  GM;CULUS. 


CHAPTER    I. 

Elementary  Methods  of  Integration. 


Integrals. 

I.  In  an  important  class  of  problems,  the  required  quanti- 
ties are  magnitudes  generated  in  given  intervals  of  time  with 
rates  which  are  either  given  in  terms  of  the  time  /,  or  are 
readily  expressed  in  terms  of  the  assumed  rate  of  some  other 
independent  variable. 

For  example,  the  velocity  of  a  freely  falling  body  is  known 
to  be  expressed  by  the  equation 

v^gt, (i) 

in  which  t  is  the  number  of  seconds  which  have  elapsed  since 
the  instant  of  rest,  and  ^  is  a  constant  which  has  been  deter- 
mined experimentally.     If  s  denotes  the  distance  of  the  body 


2  elemEaWtary  methods  of  INTEGRATIOX.    [Art.  I. 

at  the  time  /,  from  a  fixed  origin  taken  on  the  line  of  motion, 
V  is  the  rate  of  s ;  that  is, 

ds 

"  =  Tr 

hence  equation  (i)  is  equivalent  to 

'is^'     ds:-.gtdt,'..: (2) 

which  expresses  theaiiTer^ntial  of  s  in  terms  of  /  and  dt.  Now 
it  is  obvious  that  \gf  is  a  function  of  t  having  a  differential 
equal  to  the  value  of  ds  in  equation  (2) ;  and,  moreover,  since 
two  functions  which  have  the  same  differential  (and  hence  the 
same  rate)  can  differ  only  by  a  constant,  the  most  general 
expression  for  s  is 

s  =  lgt'+C, (3) 

in  which  C  denotes  an  undetermined  constant. 

2.  A  variable  thus  determined  from  its  rate  or  differential 
is  called  an  integral,  and  is  denoted  by  prefixing  to  the  given 

differential  expression  the  symbol    ,  which  is  called  the  integral 

sign*     Thus,  from  equation  (2)  we  have 


=  \gtdt, 


which  therefore  expresses  that  ^  is  a  variable  whose  differential 
is  gtdt ;  and  we  have  shown  that 


\gtdt  =  :^g^  -^  C. 


The  constant  C  is  called  the  constant  of  integration;  its 
occurrence  in  equation  (3)  is  explained  by  the  fact  that  we 
have  not  determined  the  origin  from  which  s  is  to  be  measured. 

*  The  origin  of  this  symbol,  which  is  a  modification  of  the  long  s,  will  be 
explained  hereafter.    See  Art.  100. 


I-] 


THE  DIFFERENTIAL   OF  A    CURVILINEAR  AREA. 


If  we  take  this  origin  at  the  point  occupied  by  the  body  when 
at  rest,  we  shall  have  s  =  o  when  /  =  o,  and  therefore  from 
equation  (3)  C  =0;  whence  the  equation  becomes  s  =  ^gi"^- 


The  Differential  of  a  Cu7'-vilinear  Ai'ea. 

3.  The  area  included  between  a  curve,  whose  equation  is 
given,  the  axis  of  x  and  two  ordinates  affords  an  instance  of 
the  second  case  mentioned  in  the  first  paragraph  of  Art.  i  ; 
namely,  that  in  which  the  rate  of  the  generated  quantity,  al- 
though not  given  in  terms  of  /,  can  be  readily  expressed  by  means 
of  the  assumed  rate  of  some  other 
independent  variable. 

Let  BPD  in  Fig.  i  be  the  curve 
whose  equation  is  supposed  to  be 
given  in  the  form 

y  =  f{x). 

Supposing    the    variable    ordinate 

PR  to  move  from  the  position  AB 

to   the   position    CD,  the   required 

area  ABDC\s  the  final  value  of  the  Fig.  i. 

variable  area  ABPR,  denoted   by 

y3,  which  is  generated  by  the  motion  of  the  ordinate.     The  rate 

at  which  the  area  A  is  generated  can  be  expressed  in  terms  of 

the  rate  of  the  independent  variable  x.     The   required  and  the 

dA  dx 

assumed  rates  are  denoted,  respectively,  by  —j-  and  -—  ;  an'd,  to 

dt  dt 

express  the  former  in  terms  of  the  latter,  it  is  necessary  to 
express  dA  in  terms  of  dx.  Since  x  is  an  independent  variable, 
we  may  assume  dx  to  be  constant ;  the  rate  at  which  A  is  gen- 
erated is  then  a  variable  rate,  because  PR  or  y  is  of  variable 
length,  while  moving  at  a  constant  rate  along  the  axis  of  x. 
Now  dA  is  the  increment  which  A  would  receive  in  the  time 


4  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  3. 

dt,  were  the  rate  of  A  to  become  constant  (see  Diff.  Calc, 
Art.  17).  If,  now,  at  the  instant  when  the  ordinate  passes  the 
position  PR  in  the  figure,  its  length  should  become  constant, 
the  rate  of  the  area  would  become  constant,  and  the  increment 
which  would  then  be  received  in  the  time  dt,  namely,  the 
rectangle  PQSR,  represents  dA.  Since  the  base  RS  of  this 
rectangle  is  dx,  we  have 

dA  =  ydx  =  f  {x)dx (i) 

Hence,  by  the  definition  given  in  Art.  2,  A  is  an  integral,  and 
is  denoted  by 

A  =  \f{x)dx (2) 


Definite  Integrals. 

4-.  Equation  (2)  expresses  that  yi  is  a  function  of  x,  whose 
differential  \'s,  f{x)dx  ;  this  function,  like  that  considered  in  Art. 
2,  involves  an  undetermined  constant.  In  fact,  the  expres- 
sion    f{x)dx  is  manifestly  insufficient  to  represent  precisely 

the  area  ABPR,  because  OA,  the  initial  value  of  x,  is  not  indi- 
cated. The  indefinite  character  of  this  expression  is  removed 
by  writing  this  value  as  a  subscript  to  the  integral  sign  ;  thus, 
denoting  the  initial  value  by  a,  we  write 


A 


=  \^f{x)dx, (3) 


in  which  the  subscript  is  that  value  0/ x  for  ivhicJi  the  integral 
has  the  value  zero. 

If  we  denote  the  fitial  value  of  x  {OC  in  the  figure)  by  d,  the 
area  ABDC,  which   is  a  particular  value  of  A,  is  denoted  by 


§  I-]  DEFINITE   INTEGRALS.  5 

writing   this    value    of  x    at    the    top    of   the    integral   sign, 
thus, 


ABDC 


^^f{x)dx (4) 


This   last  expression  is  called  a  definite  integral,  and  «  and 
b  are   called  its  limits.      In  contradistinction,  the  expression 

f{x)dx  is  called  an  indefinite  integral. 


5.  As  an  application  of  the  general  expressions  given  in  the 
last  two  articles,  let  the  given  curve  be  the  parabola 

Equation  (2)  becomes  in  this  case 

A  =  f  x'dx. 

Now,  since  Ix^  is  a  function  whose  differential  is  x^dx,  this 
equation  gives 

A  =  [  x^dx  =  ix^+  C, (i) 

in  which  C  is  undetermined. 

Now  let  us  suppose  the  limiting  ordinates  of  the  required 
area  to  be  those  corresponding  to  x  =  i  and  ;f  =  3.  The  vari- 
able area  of  which  we  require  a  special  value  is  now  represented 

by  [  x^dx,  which  denotes  that  value  of  the  indefinite  integral 

which  vanishes  when  x  —  i.  If  we  put  ;ir  =  i  in  the  general 
expression  in  equation  (i),  namely  ^x^  +  C,  we  have  ^  +  C; 
hence  if  we  subtract  this  quantity  from  the  general  expression, 
we  shall  have  an  expression  which  becomes  zero  when  x  =  1. 
We  thus  obtain 

A  =  f  x^dx  =ix^-i. 

Jz 


ELEMEXTARY  METHODS  OF  INTEGRATION.    [Art.  5. 


Finally,  putting,  in  this  expression  for  the  variable  area,  -r  =  3, 
we  have  for  the  required  area 


jWt- =  ^3^-1  =  81. 


6.  It  is  evident  that  the  definite  integral  obtained  by  this 
process  is  simply  tJic  difference  between  the  values  of  the  indefinite 
integral  at  the  upper  and  loivcr  limits.  This  difference  may  be 
expressed  by  attaching  the  limits  to  the  symbol  ]  affixed  to  the 
value  of  the  indefinite  integral.  Thus  the  process  given  in  the 
preceding  article  is  written  thus, 


x^dx  =  -\;v^  -\-C 


—1 1 


i  = 


The  essential  part  of  this  process  is  the  determination  of 
the  indefinite  integral  or  function  whose  differential  is  equal  to 
the  given  expression.  This  is  called  the  integration  of  the 
given  differential  expression. 

Elementary    Theorems. 

7.  A  constant  factor  may  be  transferred  from  one  side  of  the 
integral  sign  to  the  other.  In  other  words,  if  m  is  a  constant 
and  7/  a  function  of  x, 


inudx  =  vi\  udx. 


Since  each  member  of  this  equation  involves  an  arbitrary 
constant,  the  equation  only  implies  that  the  two  members  have 
the  same  differential.  The  differential  of  an  integral  is  by 
definition  the  quantity  under  the  integral  sign.  Now  the 
second  member  is  the  product  of   a  constant  by  a  variable 

factor ;  hence  its  differential  is  7nd\     u  dx   ,  that  is,  m  u  dx,  which 

is  also  the  differential  of  the  first  member. 


§  T.]  ELEMENTARY   THEOREMS.  7 

8.  This  theorem  is  useful  not  only  in  removing  constant 
factors  from  under  the  integral  sign,  but  also  in  introducing 
such  factors  when  desired.     Thus,  given  the  integral 

X"  dx ; 

recollecting  that 

^(;ir«  +  ^)  =  (;?  +  \)x"-dx, 

we  introduce  the  constant  factor  n  ■\-  \  under  the  integral  sign ; 
thus, 

\x''dx  —  — ^ —   in  +  i)x"dx  = ;r«+  '  +  C. 

9.  If  a  differential  expression  be  separated  into  parts,  its  in- 
tegral is  the  sian  of  the  integrals  of  the  several  parts.  That  is, 
if  71,  V,  w,  '  '  '  are  functions  of  x, 

\{ii-\-v-\-w-\-'''  •)dx  =  \2i  dx  +  \v  dx  +  \w dx  +  •  •  • 

For,  since  the  differential  of  a  sum  is  the  sum  of  the  differ- 
entials of  the  several  parts,  the  differential  of  the  second  mem- 
ber is  identical  with  that  of  the  first  member,  and  each  member 
involves  an  arbitrary  constant 

Thus,  for  example, 

(2  —  4/;f)  dx  =   2dx  —   x'dx  =  2X  —  ^x^  +  C , 

the  last  term  being  integrated  by  means  of  the  formula  deduced 
in  Art.  8. 

Fundamental  Integrals. 

10.  The  integrals  whose  values  are  given  below  are  called 
the  fundamental  integrals.  The  constants  of  integration  are 
generally  omitted  for  convenience. 


8  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  10. 

Formula  {a)  is  given  in  two  forms,  the  first  of  which  is  de- 
rived in  Art.  8,  while  the  second  is  simply  the  result  of  putting 
n=  —  in.  It  is  to  be  noticed  that  this  formula  gives  an  indeter- 
minate result  when  n  =  —  i  ;  but  in  this  case,  formula  {b)  may 
be  employed.* 

The  remaining  formulas  are  derived  directly  from  the  for- 
mulas for  differentiation;  except  that  (/'),  {k'),  (/'),  and   (;//') 

are  derived  from  (7),  (/-),  (/),  and  (w)  by  substituting  --  for  x. 


x"dx  = 


71  +   I 


—  =log(±;.r)t. 


a'^dx  = 


log  a 
cos^  d9  =  sin  0 


{dx_ 
Jx'"  ~ 


{in  —  1)  X" 


s^dx  =  £" 


sinede=  -  COS  ^ (c) 


*  Applying  formula  {a)  to  the  definite  integral     x"dx,  we  hav 


1: 


x"dx  = 


r  +  '-a"+' 


«  4-  I 


which  takes  the  form  -  when  «  =  —  i  ;  but,  evaluating  in  the  usual  manner, 


«  +  i       J«  =  -i 


_^''  +  'log^-a"  +  'loga 


=  log  b  —  log  a  ; 
J«  =  — I 


a  result  identical  with  that  obtained  by  employing  formula  {b). 

\  That  sign  is  to  be  employed  which  makes  the  logarithm  real.    See  Diff.  Calc. 
Art.  43. 


§!•] 


FUNDAMENTAL  INTEGRALS. 


9 

if) 

(/.) 

(0 
U) 

if) 
(/^) 
{k') 

in 
in 

{m) 


Jcos^^        J 

J  sin'^^        J 

r  sin  e  dd 
J     cos^^     ' 

J    sin-^6'     ' 

J  V(I  -  x") 
f         ^;tr 


=    sec^^^^/^  tan 


cosec^f^  dd  —  —  cot  ^  . 
sec(9  tan  6  dO  =  sec  ^ 


cosec  ^  cot^  <i^^  =  —  cosec 


sin"^  X  +  C  =  —  cos  ^  .r  +  (;7'     . 


//  9         ON   =  siii'^  -4-6^=—  cos^-+6r'.    . 


It 


dx 


dx 


—  tan"^  ^  +  (T  —  —  cot~^  ;jr  +  C. 


(     dx  I  .  X        ^  I  .  ;t:       ^ , 

-o 5  =  -tan-^-+^= cot-^-+C.    . 

ja''  +  JT      a  a  a  a 

1; 


V. 


,v  V{^  —  i) 
dx 


sec  ^  ,r  +  C  =  —  cosec  ^  ,r  +  6".   . 


//  2    — -jt  =  -  sec  ^—  +  6"= cosec"^  — b  C 

V{x*  —  a'^)       a  a  a  a 


dx 


V{2x  —  x^) 
dx 


vers  '■  X. 


-77 jr-  =  vers  ^  - 


10  ELEMENTARY  METHODS  OF  INTEGRATION.       [Ex.  I. 


Examples   I. 

Find  the  values  of  the  following  integrals  : 
[dx 


[dx 
J  x^' 

C   dx 

J"  x^ 


dx 

„5    > 


5- 


2  V^. 


2 


i 


^'xdx,  f^rl 


6.  f    (.v-i)V:^,  ^'_^>  +  ^_^. 
•''  3 

f-  0      3-8 

7.  r'(«  -  ^;c) V;c,  «V  -  abx  +  —1  ':=-'. 
Jo  3  Jo      3^^ 

8.  f"  {a  +  ;c)Vx,  a^  +  ^^  +  CA-'  +  -T  =  ^a\ 

J -a  2                                   4j-c 

9-  J^  -,  2  log  a. 

to.    J        — ,  l0g(-A-)J   =l0g2. 


I.] 


EXAMPLES. 


II 


II.    I — —ax, 


la 

r 

12.  £-^dx, 
J  o 

13.  sin  6dd, 

'  o 

14.  COsa:^.x:, 

J  o 


n4«  s. 

2  '/.v(a''  +  f«:v;  +  ^x^)  I  =  23H  •  a'\ 

-In 


ey  —  I. 


I  —  cos 


sm  -V 


i-    dd 


16. 


*■'  ^.r 


17 


dx 


i»00 
/•OO 


dx 


V  {x"  -  1) 


tan6' 


Ir'T 


19.  If  a  body  is  projected  vertically  upward,  its  velocity  after  t  units 
of  time  is  expressed  by 

a  denoting  the  initial  velocity  ;  find  the  space  Si  described  in  the  time 
ti  and  the  greatest  height  to  which  the  body  will  rise. 


u  =\  vdt  =  at^  —ig^i\ 


when  V  =  o  ,t=  — ,  s  =  — - 


12  ELEMENTARY  METHODS  OF  INTEGRATION.    [Ex.  I. 

20.  If  the  velocity  of  a  pendulum  is  expressed  by 

7Tt 

V  =  a  cos  —  , 

2T 

the  position  corresponding  to  /  =  o  being  taken  as  origin,  find  an  ex- 
pression for  its  position  s  at  the  time  /,  and  the  extreme  positive  and 
negative  values  of  s. 

2Ta      .      7Tt 

s  = sm  —  , 

n  2T 

j-  =  ± when  t  =  r,  ^T,  ^t,  etc. 


21.  Find  the  area  included  between  the  axis  of  x  and  a  branch  of 
the  curve 

y  =  sin  X.  2. 

22.  Show  that  the  area  between  the  axis  of  -v,  the  parabola 

y  =  4ax, 

and  any  ordinate  is  two  thirds  of  the  rectangle  whose  sides  are  the 
ordinate  and  the  corresponding  abscissa. 

23.  Find  (a)  the  area  included  by  the  axes,  the  curve 

J  =  ^■', 

and  the  ordinate  corresponding  to  a:  =  i,  and  (/i)  the  whole  area  be- 
tween the  curve  and  axes  on  the  left  of  the  axis  of  ^'. 

24.  Find  the  area  between  the  parabola  of  the  nih  degree, 

a"-'y  =  x", 

and  the  coordinates  of  the  point  {a,  a). 


ft  +  I 


§  I.]  EXAMPLES.  13 

25.  Show   that   the   area  between   the   axis  of  x,   the  rectangular 

hyperbola 

xy=Y, 

the  ordinate  corresponding  to  .v  =  i,  and  any  other  ordinate  is 
equivalent  to  the  Napierian  logarithm  of  the  abscissa  of  the  latter 
ordinate. 

For  this   reason   Napierian   logarithms  are   often   called  hyperbolic 
logarithms. 

26.  Find  the  whole  area  between  the  axes,  the  curve 

and  the  ordinate  for  x  =  a,  m  and  n  being  positive. 

rr     ^                        ^  ^^ 
If  n  >  m,  ; 


if  n  ^  m,  00 . 

27.  If  the  ordinate  BJ?  of  any  point  B  on  the  circle 

;t'  +_>;'=  a'  .  • 

be  produced  so  that  BJ?  •  jRF  =  a'',  prove  that  the  whole  area  between 
the  locus  of  B  and  its  asymptotes  is  double  the  area  of  the  circle. 

28.  Find  the  whole  area  between  the  axis  of  x  and  the  curve 

y  {a'  +  x')  =  a\ 

TtC^. 

29.  Find  the  area  between  the  axis  of  x  and  one  branch  of  the  com- 
panion to  the  cycloid,  the  equations  of  which  are 

x  =  atj}  y  =  a  {i  —  cos tp). 

27td\ 


14  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  II. 


II. 

Direct  Intcg7'ation. 

II.  In  any  one  of  the  formulas  of  Art.  lo,  we  may  of  course 
substitute  for  x  and  dx  any  function  of  x  and  its  differential. 
For  instance,  if  in  formula  {b)  we  put  x  —  ^  in  place  of  x,  we 
have 

J  Z^a  =  ^og  (^  -  ^)         or         log  {a  -  x), 

according  as  x  is  greater  or  less  than  a. 

When  a  given  integral  is  obviously  the  result  of  such  a  sub- 
stitution in  one  of  the  fundamental  integrals,  or  can  be  made 
to  take  this  form  by  the  introduction  of  a  constant  factor,  it  is 


said  to  be  directly  integrablc.     Thus,     sin  mxdx  is  directly  in- 

tegrable  by  formula  (c) ;  for,  if  in  this  formula  we  put  mx  for  6, 
we  have 

1 


sin  7nx  -  in dx^=  —  cos  mx , 
hence 

sin  in  X  ■  vidx  =^ cos  ni  x . 

VI 


sin  mx  dx  =  — 
J  m  J 

So  also  in  \'{a  +  bx^)  x  dx , 


the  quantity  x  dx  becomes  the  differential  of  the  binomial 
{a  +  bx"^)  when  we  introduce  the  constant  factor  2b,  hence  this 
integral  can  be  converted  into  the  result  obtained  by  putting 


{a  +  bx^)  in  place  of  ,t'in    >/  xdx,  which  is  a  case  of  formula  {a), 

lUS 

[V{a  +  bx")  X  dx  =  —\{a  +  bx'f  2bx  dx  =  -^{a  ^  bx'Y  . 


Thus 


§n.] 


DIRECT  INTEGRATION. 


15 


12.  A  simple  algebraic  or  trigonometric  transformation 
sometimes  suffices  to  render  an  expression  directly  integrable, 
or  to  separate  it  into  directly  integrable  parts.  Thus,  since 
—  sin  X  dx  is  the  differential  of  cos  x,  we  have  by  formula  {U) 

sin  x  dx 


tan  X  dx  ^ 


cos  X 


=  —  log  cos  X  . 


So  also,  by  formula  (/), 


han^  Odd  = 
by  (e)  and  (a), 


{sec^d-  i)dd=tsine  -d; 


|sin=*  edd  -- 
by  (7)  and  {a), 


(I  -  cos^  d)  sin  ddd=-cosd  +  i^  cos^  d ; 


/(f^V-^  =  l7fr^)^^ 


dx 


V  {i  —  ^)      2 . 


(i  -  ;j^)-*  (-  2x  dx)  =  sin-^  x-  V  {i  -  x^). 


Rational  F^^actions. 

13.  When  the  coefficient  of  dx  in  an  integral  is  a  fraction 
whose  terms  are  rational  functions  of  x,  the  integral  may  gen- 
erally be  separated  into  parts  directly  integrable.  If  the  de- 
nominator is  of  the  first  degree,  we  proceed  as  in  the  following 
example. 

r,v^  —  ;f  +  3 


Given  the  integral 
by  division, 


2X  -^   \ 

x^  —  X  +  ^  _x      3 
2x  +  1     ~  2      4 


dx; 


+ 


15       I 

42^+1 


1 6  ELEMENTARY  METHODS  OF  INTEGRATION,  [Art.  I3, 

hence 

f'^  -  ^  +  3  V.  _  ^  f  w.  _  3  f  ..  ^  15  f     dx 


dx  =  ]\xdx-^[dx  +^  [ 


2  ,V  +   I  2  J  4  J  4  J  2X  +  I 

X'  7.x  IK  ,  ,  . 

44s 

When  the  denominator  is  of  higher  degree,  it  is  evident  that 
we  may,  by  division,  make  the  integration  depend  upon  that  of 
a  fraction  in  which  the  degree  of  the  numerator  is  lower  than 
that  of  the  denominator  by  at  least  a  unit.  We  shall  consider 
therefore  fractions  of  this  form  only. 

/denominators  of  the  Second  Degree. 

14.  If  the  denominator  is  of  the  second  degree,  it  will  (after 
removing  a  constant,  if  necessary)  either  be  the  square  of  an 
expression  of  the  first  degree,  or  else  such  a  square  increased 
or  diminished  by  a  constant.  As  an  example  of  the  first  case, 
let  us  take 

The  fraction  may  be  decomposed  thus : 

,r  +  I         A-  —  I  +  2  I 


+ 


{x-\f-    {x-\J    -  x-\^  {x-\f' 
hence 


[   x  +  \     J        {    dx  [      dx 


2 


=  log  (x—  l)  — 


16.  The  integral  f    ./'  "^  ^     .  dx 

^  }  X-  +  2x  +  6 


§11.]  DENOMINATORS  OF   THE   SECOND  DEGREE.  1 7 

affords  an  example  of  the  second  case,  for  the  denominator 
may  be  written  in  the  form 

x^  +  2x  +  6  —  {.V  +1)^  +  5. 

Decomposing  the  fraction  as.  in  the  preceding  article, 

;r  +  3  X  +  I  2 


{x+  if+s      Gt'  +  i)'+  5      ('t-+i)'+5' 
whence 

^  +  3      .^^^  _  f  (^  +  I)  ^'^'    ^  o  f         ^-^' 


nx+  i)dx    ^  J d^ 

](x  -{-  if  +  5         Ju'  +  i)^ 


x^  -\-  2x  +  6   '       J (-1^  +  i)^  +  5        ]{x  +  if  +  S' 

The  first  of  the  integrals  in  the  second  member  is  directly 
integrable  by  formula  (b),  since  the  differential  of  the  denom- 
inator is  2  {x  +  i)dx,  and  the  second  is  a  case  of  formula  {k'). 
Therefore 

'       -^  ■    3       dx  =  4-  log  x.^^  +  2x  +  6)  +  -—  tan"^ — 


ix"  +  2x  +  6   '        -     ^  ^'     '     '    '     ^   ■    4/5  V5 

16.  To  illustrate  the  third  case,  let  us  take 

2x  +  I 


j^- 


dx, 


in  which  the  denominator  is  equivalent  to  {x  —  \f  —  6|-,  and 
can  therefore  be  resolved  into  real  factors  of  the  first  degree. 
We  can  then  decompose  the  fraction  into  fractions  having  these 
factors  for  denominators.  Thus,  in  the  present  example,  as- 
sume 

2^+1  A  B  ,  ^ 

"r-—-,      ....     (I) 


Pi?'  —  X  —^        X  —  ^        X  +  2 

in  which  A  and  B  are  numerical  quantities  to  be  determined. 
Multiplying  by  {x  -  3)  {x  +  2), 

2x  +  1  =  A{x  +  2)  +  B{x  —s) (2) 


1 8  ELEMENTARY  METHODS  OF  INTEGRATION.  [Art.  l6. 

Since  equation  (2)  is  an  algebraic  identity,  we  may  in  it  assign 
any  value  we  choose  to  x.     Putting  x  =  3,  we  find 

y  =  $A,  whence  A  =  ^, 

putting  X  =  —  2, 

—  3  =  —  5^,        whence  B  =  ^. 

Substituting  these  values  in  (i), 

2;tr  +  I      _  7         ^  3 

,i^-x-6      5(-^-3)      SU  +  2)' 
whence 

l^^l^le"^"-'  =  ^  ^°-  (-"  -  3)  +  *  ^°g  ("^  +  -)• 

17.  If  the  denominator,  in  a  case  of  the  kind  last  considered, 
is  denoted  by  {x  —  a)  {x  —  b),  a  and  b  are  evidently  the  roots  of 
the  equation  formed  by  putting  this  denominator  equal  to  zero. 
The  cases  considered  in  Art.  14  and  Art.  15  are  respectively 
those  in  which  the  roots  of  this  equation  are  equal,  and  those 
in  which  the  roots  are  imaginary.  When  the  roots  are  real  and 
unequal,  if  the  numerator  does  not  contain  x,  the  integral  can 
be  reduced  to  the  form 

f  dx 

]{x-a){x-by 

and  by  the  method  given  in  the  preceding  article  we  find 

f  dx  I 


[x  —  a)  [x  —  b)      a  —  b 


log  {x  -a)  -  log  {x  -  b) 


'    log-^" (Ay 


a-b'^^  x-y 


*  The  formulas  of  this  series  are  collected  together  at  the  end  of  Chapter  II., 
for  convenience  of  reference.     See  Art.  loi. 


§  11.]         DENOMWATORS  OF  THE  SECOND  DEGREE.  I9 

in  which,  when  x  <  a,  log  {a  —  x)  should  be  written  in  place  of 
log  {x  —  a).     [See  note  on  formula  {b),  Art.  10.] 
If  ^  =  —  a,  this  formula  becomes 

{     dx  I    .      X  ~  a  ,  ... 


ix^  —  a^      2a         X  +  a 

Integrals  of  the  special  forms  given  in  (A)  and  (A')  may  be 
evaluated  by  the  direct  application  of  these  formulas.  Thus, 
given  the  integral 

f  ^^ 

l2x^  +  ^x  ~  2' 

if  we  place  the  denominator  equal  to  zero,  we  have  the  roots 
«  =  1.,  ^  =  —  2;  whence  by  formula  (A), 

r  dx  _jf  ^^  _^^i-^~i. 

J  2^  +  2,^  —  2~  '^]{x  —  -|)  (x  +  2)  ""■  2  '  ij   °^  X  +  2 ' 

or,  since  log  (zx  —  i)  differs  from  log  {x  —  D  only  by  a  con- 
stant, we  may  write 

f  dx  I  ,       2,r  —  I 

=  -  lof 


hx^  +  ^x  —  2      5     ^   X  +  2 


Denominators  of  Higher  Degree, 

18.  When  the  denominator  is  of  a  degree  higher  than  the 
second,  we  may  in  like  manner  suppose  it  resolved  into  factors 
corresponding  to  the  roots  of  the  equation  formed  by  placing  it 
equal  to  zero.  The  fraction  (of  which  we  suppose  the  numerator 
to  be  lower  in  degree  than  the  denominator)  may  now  be  decom- 
posed into  partial  fractions.  If  the  roots  are  all  real  and  un- 
equal, we  assume  these  partial  fractions  as  in  Art.  16;  there 
being  one  assumed  fraction  for  each  factor. 

If,  however,  a  pair  of  imaginary  roots  occurs,  the  factor  cor- 


20  ELEMENTARY  METHODS   OF  INTEGRATION.  [Art.  1 8. 

responding  to  the  pair  is  of  the  form  (,r  —  af  +  /^,  and  the 
partial  fraction  must  be  assumed  in  the  form 

Ax  ^  B 


{x-af  +  fS'^' 


for  we  are  only  entitled  to  assume  that  the  numerator  of  each 
partial  fraction  is  lower  in  degree  than  its  denominator  (other- 
wise the  given  fraction  which  is  the  sum  of  the  partial  fractions 
would  not  have  this  property). 


19.  For  example,  given 


^  +  3  ^^. 


)(x'+  l)ix-  I) 
Assume 


X  +  3  Ax  -^^  B         C  ,  . 


(.r^  +  I)  (^  —  i)        x^  +  I        X  —  I 
whence 

X  +  3  =  {x-  i)  {Ax  +  B)  +  {x^+  i)  C. 
Putting  x=  1, 

4  =  2(7,  whence       C  =  2; 

putting  X  =  o,  • 

3  =  —  B  +  C,      whence       B  —  —  i. 

To  determine  A,  any  convenient  third  value  may  be  given 
to  x;  for  example,  if  we  put  x  =  —  i,  we  have 

2  =  ~2{- A  +  B)  +  2C  .-.  A=-2. 

Substituting  in  (i\ 

,r  +  3  _      2  2.y  +  I 

{x^ -^  l){x —l)'~  X  —  I        ;ir2  +  I  ' 


§11]- 


DENOMINATORS  OF  HIGHER  DEGREE. 


21 


therefore 


J(,t-2+i)(;i;-l)^-^~4r- 


■  2x  dx 


dx 


,r  +  I 


=  2  log  {x  —  i)  —  log  (,1-^  +  I)  —  tan   ^  X. 

20.  If  the  denominator  admits  of  factors  which  are  func- 
tions of  ,r^,  and  the  numerator  is  also  a  function  of  ,r^,  we  may 
with  advantage  first  decompose  into  fractions  having  these 
factors  for  denominators.     Thus,  given 

f  x^dx 
Putting  J  for  x^  in  the  fraction,  we  first  find 


y     _ 


+ 


f  —  a^      2(j  +  c^)      2(  J  —  c^) 


hence 


x^dx 


j^  —  cc 


1  —  "S   \  Ji 


dx 


+ 


dx 


x^  -\-.a" 


therefore  [see  equation  {A'),  Art.  17], 


x^dx         I    ,       X  —  a        I  .X 

—x 1  =  —  log 1 tan~  ^  - . 

XT  —  a*      4a         xj  +  a      2a  a 


This  method  may  sometimes  be  employed  when  the  nume- 
rator is  not  a  function  of  x^ ;  thus,  since 


x'  -  a'      2^(,r3  -  a^)      2a^{x^  +  a^ 


we  have 


x 


X 


X^~a^      2d^  {x^  -  a')      2a^  {x^  +  a^) ' 


22  ELEMENTARY  METHODS  OF  INTEGRA  TIOX.    [Art.  20. 


hence 


xdx  I    .       x^  —  (^ 

=  7:5  ^og 


Jx*  —  a*     40^     ^  x^  +  a^' 

21.  The  fraction  corresponding  to  a  pair  of  equal  roots,  that 
is,  to  a  factor  in  the  denominator  of  the  form  {x  —  a)\  is  (see 
Art.  14)  equivalent  to  a  pair  of  fractions  of  the  form 

A  B 

+ 


X  —  a      \x  —  ay 

we  may,  therefore,  at  once  assume  the  partial  fractions  in  this 
form.  We  proceed  in  like  manner  when  a  higher  power  of  a 
linear  factor  occurs.     For  example,  given 

f  x  ^  2  , 

ax\ 


]{x  -  \f{x  +  i) 
we  assume 


X  -^  2  A  B        ^      C  D 

+  7 To  + :  + 


{x  -  l)\x  +  1)      {x-  if  ^  {x  -  1)2  "  .1-  -  I      .V  +  r 
whence 

x  +  2=[A  +  B{x-  i)^C{x-if]{x  +  i)  +  D{x-i)\    .  (I) 
Putting  X  —  I,  we  have 

3  =  2yi         .-.         A=l. 

The  values  of  B  and  C  may  be  determined  as  follows :  if  we 
substitute  the  value  just  determined  for  A,  equation  (i),  is 
identically  satisfied  by  a-  =  i,  hence  it  may  be  divided  by  x—  i. 
We  thus  obtain 

-\=iB  ^  C{x-  i)-]{x+  i)+D{x-if  .     .     (2) 


§  II.]  MULTIPLE  ROOTS.  23 

in  which  we  may  again  put  ;r  =  i,  whence  B  =  —  \,     In  like 
manner  from  (2),  we  obtain 

\  =  C{x^\)+D{x-\), 

from  which  C  =^ ,  and  D  =  —\.     Therefore 

dx 


•^+^         dx^l. 


(;ir— l)^(-r+l)  2]{x—\f       Af]{x—\) 


'    dx  \  {    dx  \{  dx    _\ 

(x- 1)^  ~  4  J  (x-  lY  "^  sji^       8 


X  +  I 


3  I  I  -       ;ir  —  I 

4{x-if^4{x-  I)  ^8     ''X+  I 

22.  In  this  example,  after  obtaining  the  values  of  A  and  D 
from  equation  (i)  by  putting  x  =  i,  and  x  =  —  i,  two  equations 
from  which  B  and  C  might  be  obtained  by  elimination  could 
have  been  derived  by  giving  to  x  any  two  other  values.  Con- 
venient equations  for  determining  B  and  C  may  also  be  obtained 
by  putting  ;t:  =  i  in  two  equations  successively  derived  by 
differentiation  from  the  identical  equation  (i).  In  the  first  dif- 
ferentiation we  may  reject  all  terms  containing  (x  —  i)^;  since 
these  terms,  and  also  those  derived  from  them  by  the  second 
differentiation,  will  vanish  when  x  =  i.  Thus,  from  equation 
(i),  Art.  21,  we  obtain 

1  =  A  +  2Bx  +  2C  (x^  —i)  +  terms  containing  (x  —  ly. 

Putting  X  =  I,  and  yi  =  | ,  we  have  B  =  —  I.     Differentiating 
again  and  substituting  the  value  of  B, 

o  =  —  I  +  4CX  +  terms  containing  {x  —  i), 
and,  putting  ;ir  =  i  in  this  last  equation,  C=  ^ , 

23.  When  the  method  of  differentiation  is  applied  to  a  case 


24  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  23. 

in  which  more  than  one  multiple  root  occurs,  it  is  best  to  pro- 
ceed with  each  root  separately.     Thus  given, 


f :L±J dx 


;tr+  I  A  B  C  D 

+      +7 T-I^^  + 


{X  -  \f  {X  +  2)2         {X-    if'      X-l  (X  +   2)2    '    X  +  2 

whence 

x+i=[A+B{x-i)]{x  +  2f+[C+D{x  +  2)]{x-iy  .   .    (i) 
Putting  X  =  I,  and  x  =  —  2,  we  derive 

A  =  '-,  C=-'-. 

9  9 

Differentiating  (i),  we  have 

I  =  2A  (x  +  2)  +  B  (x  +  2)2  +  terms  containing  {x  —  i), 

2  I 

whence,  putting  x  =:  i,  and  A  =^  -,  we  have  B  = . 

Again,  differentiating  (i),  we  have 

1  =  2C  {x  —  i)  +  D  (x  —  if  +  terms  containing  {x  +  2), 
whence,  putting  x  =  —  2,  and  C  =  —  ,  we  have  D  =  — . 

Therefore 

f  -*'  +  I  ^    —  _         2  I  J^ ,      x  +  2 

](x-if{x  +  2f  '''~       9  {x  -  I)  ^  9  (a-  +  2)      27  °^;»r-l  ' 

24-.  Instead  of  assuming  the  partial  fractions  with  undeter- 


II.] 


RATIONAL  FRACTIONS. 


25 


mined  numerators,  it  is  sometimes  possible  to  proceed  more 
expeditiously  as  in  the  following  examples : 
Given 

dx ; 


\x^  (I  ■ 


putting  the  numerator  in  the  form  i  -^  x^  —  x^,  we  have 


I         ^    _f_l  +  f!_^    _f__f!__ 


dx 


J  a-^       J  -r  ( I  +  x^\ 


(i+x^) 
Treating  the  last  integral  in  like  manner, 


(dx       (dx       f  X  dx 


4-   locr  — !— : 

2X?  ^     ^  X 


=  -  ^  -  log  ^  +  i  log  (I  +  ,r2)  = 
Again,  given 

]x^{\  +  ,r) 
putting  the  numerator  in  the  form  (i  +  xf  —  2x  —  x?-,  we  have 
r         I  ^    —  W^      f    2  +  ;r       , 

Hence  by  equation  (^),  Art.  17, 


(^;ir 


.r       I  +  ^ 


26  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II, 


■I 


Examples  II. 

dx 


r_  dx 

J  {a-xY' 
[   xdx 

J  o 

8.       («  +  w^)'  dXy 
0.      cos'  X  sin  A  dXy 

■I 


fcos  0  //9 


sin  0 


—  log  (^  -  x\ 

I 

a  —  -v 

\  log  (a'  +  A-). 

^3  -  (a^  _  ^)l 

3 

I,          a' 

a  -  i/{a'  -  .v^). 

(a^  +  3-ry 

24 

(^  4-  //u-y—a" 

Zm 

cct  2^ 

I  —  COS   X 


''•        „•  s,.  "  >  cosec  9 


2 


12.    I  sec'  3  A-  tan  3-v  ^,  sec^jjc 1 

Jo  O 


§n.i 


EXAMPLES. 


27 


17.  COS 


13,  a'"^dx, 

14.  (f-^—  i)Vjf, 

^"-  {a  —  x)  dx 
o  Vi^ax  —  x') ' 

r 

J  o 

18.  sec*0^e, 

19.  tan'^rt'^, 

20.  sec^  .^  tan  .r  ^ji:, 

J  o 

21.  4/ ^JV, 

It 

22.  ""  cot' 0^0, 
4 

24.     sin  (a  —  2e)  aTe, 


w  log  a ' 

(i  +  3sin'jt:)'' 
't/(2aA:  —  a;°)        =  o. 


tan  9  +  -  tan'  9. 
3 


-tan^'jt:  4-  log  cos  X 


It 

^sec^^l^:..^. 
4  Jo        4 


a  sin~^  -  +  Via"  —  x"). 
a 


1  —  log  2 


i^iaax  —  x^^  +  a  vers   ^  —  0 


cos(«—  29) 
2 


28  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 


25- 


26. 


27. 


28. 


29. 


30- 


31- 


32. 


ZZ- 


34- 


35- 


cos  X  dx 


a  —  b%\xix'' 

4    dx 
^  tan  X  ' 

6 

tan  -v ' 
dx 


1  -V  log  X 


dx 


£^  +  e- 


x'  dx 


A-"    +    I  ' 

X  dx 

Via'  -  X*) ' 

dx 

V(5-3-v')' 

dx 

2  +  5-v- ' 

dx 

...   . 

^.vy(2A--l)' 
P   +   -V  +    I  ' 


— -7log(^  —  ^sin.v). 


i  log  2. 


log  (-  log.v) 


\  log  2. 


=   -  log  2. 


tan-  'f'. 


3 

tan  "  '-v' 

1 
2 

sin     -5- 

t 

^3 

sin 

V5 

I 

Vio 

tan" 

V2 

4 

Vz^'"''  VZ    1~3^3" 


§n.] 


EXAMPLES. 


29 


36. 


37- 


dx 


„  4/(5  -^x-  x^y 


COS    ^  f . 


r  i/(^v^  -  a') 


dx 


L~J-vV 


3  2 

.r"  —  «' 


-/(a    —  a  ) 


39- 


40.-   -  .r  +  3 


[■40:    —  X  - 
J       a-"^  +  I 


dx^ 


a'  (log  2  -  f ). 
4^;  —  ^  log  (^'  +  i)  —  tan  "  ^ X 


40.     -^ ^r,    AT  +  log  (.r''  —  X  +  I)  -\ T-tan  '^ ; — 


41.    [3 ^-dx. 


42. 


43- 


44. 


.V    -  4 

(i  +^r 


.r  —  AT 


^, 


3  ,      -r  —  2 

.a;  +-l0g ; 

4  °:v  +  2 


log 


^(2.y  +  i)'  dx 

2X  +   3  ' 

r    2Jr  +  3 


dx^ 


45 


(2:1:  +  i) 

3^^"  +  3 


(I  -  xy      ■ 

X^  —  X  +  2  log  {2X  +  3). 


-  log  {2X  +1) ^ 

2       °  ^  2a-  +    I 


•  J.r  -  3a; 


46. 


+  2 


^;f, 


a:  +  log 


X  —  2 

X  —  I 


X  —2ax  cos  a  ■\-  a 


I  ,  jv  —  a  cos  a~l'*        ^  —  '^ 

'—. tan  ■  * -. = -. —  . 

^  sin  a:  a  szn  a  2a  sin  ot 

— 'o 


30  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 


47 


dx 


2ax  sec  a  +  a 
dx 


f       3-v  —  I 
50.        .  -^     , dx, 

*^         J  -V    —  A*  —  2X        ' 


51 


^r^il- 


(-v+2)(Ar+3)*' 
[  xdx 

]x'  +  X^-2' 


53 


f  jr'  —  .r  +  2    . 


55 


^a: 


'  -  ;c'  -  a:  +  I ' 

r     ^ 


57 


2a  tan  a 


log 


X  - 

—  <z  sec  a  — 

a  tan  or 

^x- 

-  a  sec  a  + 

a  tan  a ' 

V2 

2-r  -  2 

-34/2 

12 

^°S  2a-  -  2 

+  3  4^2" 

^log 

14-^ 

l-X*' 

6^°S    (-V 

-2)' 

2 

log't'- 
^  X  +  2 

3 

'x  +  3' 

-.-> 

r  +  In.  ^(-^' 

•  +  1)1 

I  ,      jf  —  I        4/2  ,     :r 

-log—-—  +-^— tan-'-7- 
6     °jf  4-  I  3  V2 


2  .      j;  4-  I       I  ,      :r  —  2 


£.       Jf  4-  I I 

4      °JC  —  I         2(a:  —  l)" 


,      {x—  i)  V{x  4-1)      I        _, 

log  -^^ — ^^ — 5 tan     X. 

^         {x'  4-  i)'  2 


^^+_£)L+_I_tan->?^^. 
V3  Vs 


6^0g^^_;,+     , 


58-  J(,_,)^(,^.,,^>     -log(^-i)--log(.+x)-^^^-^^. 


*l. 


^X'-I^J  /_,i  ~"^/v-v' 


k3 


i^'lnLii^;.i.t->t   ''<'r'     C'i:^ifi^(£A\UL    ^i^*     JC 


09 


§11.] 


EXAMPLES. 


31 


f dx  

\ogx log  (i  +  df) log  (i  -^  x^) tan  -  '  x. 


1  x^  —  x  -{■  1 

2  °^a:"  +  ^  +  I 


6o-      -4—; — 5—; —  "^> 
J  :x;    +  :V    +  I 

61.  I  ^^f  ~.'  dx,       JlogJc  + -log(^- 2)  +^log(j;  +  3) 
]  X   -V  X  —  tx  6  2  3 

62.  -^ 1 , 

]  X    —  X    —  \2 


-log 1 ^tan-'-^-. 

7     ^  a:  +  2  7  4/3 


^3-  J(p-:r^- 


64. 

65- 

66. 


^-"-    -  3^  ^^ 


x'  -x"  -2' 

dx 
{x^  +  a^)  {x^  by      b""  + 


1        x—i X 

4     ^a:  +  I        2{x'  —  i) 


^-  tan"  ' log 

2a  a       4a        jc  +  a 


I ,      a;  —  2 

6^°s^^TT 


I    r,      ^  +  3     ,  ^  ^  - 1  •^n 


^7-  L(ZT^ 


^)(x^  +  by 


68 


d'dx 


2ab  {a  +  b)' 
4  —  TT 


69.     —7 — -^rr^, 

Ja:  (i  +  x') 

[        dx 


4a 


tan-^.v  +  log 


V{i  +  ^•') 


70 


I        I 

tan  - '  ;v  H 1 

X      s^ 


32  ELEMENTARY  METHODS  OF  INTEGRATION.      [Ex.  II. 


71- 


72. 


dx 


.v(i+.rr' 
dx 


log— + 


X  {a  +  bx') ' 


I   +  A-  I   +  A- 


I    ,  A-' 

—  log 


3a        a  +  ^-v' ' 


I      ,     b  .     a  •\-  b^ 
3  +  — a  log i — 


74.  Find  the  whole  area  enclosed  by  both  loops  of  the  curve 

y-A-'(i-a').  f 

75.  Find  the  area  enclosed  between  the  asymptote  corresponding 
to  -V  =  a,  and  the  curve 


,.2  ,,»     ■_   ^a..a  ^' ,,' 


x'f  +  a'x'  —  a'f 

76.  Find  the  whole  area  enclosed  by  the  curve 

'    d'y"  =  x'  {a-  -  A-'). 

77.  Find  the  area  enclosed  by  the  catenary 


2a . 


W. 


r—    X  j:— 1 


the  axes  and  any  ordinate. 


p-r']. 


78.  Find  the  whole  area  between  the  witch 
xy^  =  40*  {2a  —  x) 
and  its  asymptote.     See  Ex.  23. 


^na. 


§111.] 


TRIGONOMETRIC  INTEGRALS. 


33 


III. 

Trigono7netric  Integrals. 

25.    The    transformation,     tan'"^  6*  =  sec^  ^  —  i ,    suffices    to 
separate  all  integrals  of  the  form 

tan«^^^, (I) 

in  which  n  is  an  integer,  into  directly  integrable  parts.     Thus, 
for  example, 

[tan^  edB  =  [tan^  B  (sec^  ^  -  \)  dQ 

tan*  B       W     ^  a  jQ 
tan^  6  ad. 


Transforming  the  last  integral  in  like  manner,  we  have 

f      ^  ^  ,^      tan*  B      tan^  B      [^       .   ,^ 

\\.d.n^  B dB  T^i  ■ •+   tan<9^6'; 

J  4  2  J  ' 

hence  (see  Art.  12) 


f      .-  ,  ,„      tan*^      tan2  6'  '  „   " 

tan^  BdB  = log  cos  B. 

J  42 

When  the  value  of  n  in  (i)  is  even,  the  value  of  the  final  inte- 
gral will  be  B.     When  n  is  negative,  the  integral  takes  the  form 

[cot«  B  dB, 
which  may  be  treated  in  a  similar  manner. 


34  ELEMENTARY  METHODS  OF  INTEGRATION.     [Art.  2^. 

26.  Integrals  of  the  form 

[seC^^^ (2) 

are  readily  evaluated  whcji  n  is  an  even  number^  thus 
U&z^dde  =  f(tan2  +  i)2  see  ddS 

=  [tan*  dsG(?ddd+2  ftan'  6  sed^Sdd  +  [see*  6  dd 

tan''  6      2  tan^  6 

=  — - —  + +  tan  6. 

5  3 

If  n  in  expression  (2)  is  odd,  the  method  to  be  explained  in 
Section  VI  is  required. 

Integrals  of  the  form     cosQCOdd  are  treated  in  like  manner. 

Cases  in  which  sin'"  d  cos"  d  dd  is  directly  integrable. 

27.  If  ;^  is  2.  positive  odd  number,  an  integral  of  the  form 

[sin'«  d  cos"  QdQ (3) 

is  directly  integrable  in  terms  of  sin  6.     Thus, 


sin2  ^  cos' ^  ^^  = 


sin' 6^  (I  -sm^df  cos  ddB 


_  sin^  6      2  sin'  6      s\n^  9 
~~3  V~^      7 

This  method  is  evidently  applicable  even  when  m  is  frac- 
tional or  negative.     Thus,  putting  7  for  sin  d, 


§  III.]  TRIGONOMETRIC  INTEGRALS.  35 

fcos^^  (•(!  -f)dy{    _^  r    . 

hence 

f  cos^  e       __  ^       2    a_       2      3  +  sin2^ 

Jsint  ^"^^  --^y  '       3-^'   ~  ~  3  '     i/(sin^)- 

When  7;^  in  expression  (3)  is  a  positive  odd  number,  the  in- 
tegral is  evaluated  in  a  similar  manner. 

28.  An  integral  of  the  form  (3)  is  also  directly  integrable 
when  m  +  n  w  a7i  even  negative  integer,  in  other  words,  when  it 
can  be  written  in  the  form 


rsin'"  d  do 
cos"'+^^  6 


tan'"  S  sec^?  B  dB, 


in  which  q  is  positive. 
For  example, 

dB 


hence 


-r^^ —^  =  (tan  Byl  sec*  B  dB 

Jsina  B  COS3  ^     J^         ^ 

=  [(tan  B)--2  (tan^  ^  +  i)  sec^  ^^^ ; 

c         dB  2       ^  2 

.  3.  n ^~7i  —  T  tans  S  —  - — T-.  • 

jsm-i  B  cos-^' B      3  tan^  ^ 


It  may  be  more  convenient  to  express  the  integral  in  terms 
of  cot  B  and  cosec  B,  thus 

j^^l^  =|cot*  B  (cot2  ^  +  I)  cosQc"  BdB 
cot''  B      cot^  B 


36  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  28. 

Integrals  of  the  forms  treated  in  Art.  25  and  Art.  26  are  in- 
cluded in  the  general  form  (3),  Art.  27.  Except  in  the  cases 
already  considered,  and  in  the  special  cases  given  below,  the 
method  of  reduction  given  in  Section  VI  is  required  in  the 
evaluation  of  integrals  of  this  form. 


The  Integrals  sin2  e  dd,  and  \  cos^  e  do. 

29.  These  integrals  are  readily  evaluated  by  means  of  the 
transformations 

sin^^  =  ^(i  —  cos  26'),       and       cos^6' =  ^(i  +  cos  2^). 
Thus 

[  sin^  edB  =  I  [dd  -  I  [cos  26  dd  =  ^6  -  \s\n2e, 

or,  since  sin  26  =  2  sin  6  cos  0, 

[ sin^  ddd  =  \{d  -  sin  d  cos  6) {B) 

In  like  manner 

[cos2^^^=^(^  +  sin(9cos^) [C) 

Since  sin^  d  +  cos^  d  =  i,  the  sum  of  these  integrals  is  \d6', 


ac- 


cordingly we  find  the  sum  of  their  values  to  be  6. 

In  the  applications  of  the  Integral  Calculus,  these  integrals 
frequently  occur  with  the  limits  o  and  ^n  ;  from  {B)  and  {C) 
we  derive 


['  sin^d de  =  ['  co^ Odd  =  \7t. 


III.] 


TRIGONOMETRIC  INTEGRALS. 


37 


The  Integrals 
30.  We  have 


do 


sin  e'cos 


do 
sin  6 


,  and 


r  de 


cos  B 


Csec^edd       , 

•    /i        /I  =    -^T TT-  =  log  tan  6'. 

sin  6/  cos  6^     J     tan  (9  ^ 


Again,  using  the  transformation, 

sin  ^  =  2  sin  }d  cos  |^, 


(^) 


we  have 


hence 


f  ^^   _r        jdd         _f 
J  sin  ^^  J  sin  ^6*  cos  A/9  ~J 


•secneud 


tani(9 


j^,  =  logtani^ (^) 


This  integral  may  also  be  evaluated  thus, 


J  sin  (9  ~  J 


sin  Odd 


sirr 


sin  d  dd 


I  —  cos^  6 ' 


Since  sin  B  dd  —  —  d{cos  B),  the  value  of  the  last  integral  is,  by 
formula  (^'),  Art.  17, 


I ,      I  —  cos 

—  log 

2  I    +  cos 


=  log|/l 


I   —  COS 


+  COS 


and,  multiplying  both  terms  of  the  fraction  by  i  —  cos  B,  we 
have 


dB        ,       I  —  cos 

7-  =  log : TT- 

sm  ^         '='      sm  B 


{E) 


38 


ELEMENTARY  METHODS  OF  INTEGRATION:     [Art.  3 1. 


31.  Since  cos  d  =  sin  {\7C  +  6),  we  derive  from  formula  {E), 

By  employing  a  process  similar  to  that  used  in  deriving  for- 
mula {£'),  we  have  also 


dd         .      I  +  sm  ^ 

n   =  lOR 7{ • 

cos  U  °        COS  U 


{F') 


Miscellaneotis   Trigonometric  Integrals. 

32.  A  trigonometric  integral  may  sometimes  be  reduced, 
by  means  of  the  formulas  for  trigonometric  transformation,  to 
one  of  the  forms  integrated  in  the  preceding  articles.  For 
example,  let  us  take  the  integral 

f  dd 


]as>md  -\-  b  cos  6' 
a  =  k  cos  a,  b  =  k  sin  a, 


I  r 


dd 


Putting 
we  have 

f  rfft  T      . 

J ^  sin  6*  +  ^  cos  6~  k  ]sm{d  +  a)' 
Hence  by  formula  {E) 

f  dd  I  1      .       1  //I        \ 

; =  -  log  tan  -id  +  a)) 


or,  since  equations  (i)  give 

k  =  V(a2  +  ^), 

f  dO 


tan  a  =  —, 
a 


(I) 


^^  locr  \_2^x\  — 

as\nO  +  b  cos  0      V{a^  +  b^)    "^         2 


+  tan  -  ' 


-1 


§  III.]    MISCELLANEOUS   TRIGONOMETRIC  INTEGRALS.  39 

33.  The  expression  sin  md  sin  nd  dd  may  be  integrated  by 
means  of  the  formula 

cos  {7n  —  n)  d  —  cos  {m  +  7i)  6  =  2  sin  md  sin  n6  ; 

whence 

f  .        ^   .      ^  r^      sin  (m  —  Jt)  6     sin  (;;?  +  ;«)  (9  ,  . 

J  2  {pz  —  71)  2  {m  +  n) 

In  like  manner,  from 

cos  {m  —  n)  6  +  cos  {m  +  n)d  —  2  cos  md  cos  ;«6', 

we  derive 

f  fl  ^  ^/3      sin  {m  -n)d      sin  (;;^  +  7i)  0  .  . 

\cosm9cosnd  ad  = 7 ^ 1 — -—, ; — r— •    •    {2) 

J  2  (;;z  —  fi)  2  {m  +  n) 

When  m  =  n,  the  first  term  of  the  second  member  of  each 

of  these  equations  takes  an  indeterminate  form.     Evaluating 

this  term,  we  have 

{  ■  1    a  jQ      ^      sin  27i9  f  . 

\sm^7iddd  = , (3) 

J  2  4« 


\cos^  lid  dd  =  - ^ '     (4) 

J  2         An 


o    ^  ,^      d      sm  2nd 
and 

4« 


Using  the  limits  o  and  tt  we  have,  from  (i)  and  (2),  whe7i  m 
and  n  are  unequal  integers, 

•IT  fir 

sin  77id  sin  nd  dd  =      cos  md  cos  nd  dd  =  o;  .    .    (5) 

Jo  Jo 

but,  when  7n  and  7i  are  equal  t7itegers,  we  have  from  (3)  and  (4) 
r  sin"  7id  dd  =  r  cos2  fiddd  =- (6) 

Jo  Jo 

34.  To  integrate  4/(1  +  cos  ^)  dd,  we  use  the  formula 
2  cos''  ^^  =  I  +  cos  6", 


40  ELEMENTARY  METHODS  OF  INTEGRATION.    [Art.  34. 

whence  V(i  +  cos^)  =  ±  4/2  cos^^, 

in  which  the  positive  sign  is  to  be  taken,  provided  the  value  of 
B  is  between  o  and  n.     Supposing  this  to  be  the  case,  we  have 

f  V(i  +  Qosd)de=  V2  [cosi^^^ 

=  24/2  sin  \d. 
For  example,  we  have  the  definite  integral 


I 


4/(1  +  cos  6)  dB  —  24/2  sin  -  =  2. 

4 


Integration  of -, -r,' 

*  -^   a  ^  0  cos  6* 

'35.  By  means  of  the  formulas 

I  =  cos'^  ^^  +  sinH^       and       cos^  =  cosU^  —  sin'^^^, 
we  have 

f       dd         _  f de_ 

Jrt  +  ^cos^~  J(^  +  /;)cosH6'  ^  {a-b)  sin^^' 

Multiplying  numerator  and  denominator  by  sec^|^,  this  be- 
comes 

sec^^^rt'^ 


U  +  ^  +  (rt-^)tanH^' 
and,  putting  for  abbreviation 

tan  \e  =  yy 
we  have,  since  \  sed^^ddd  =  dy, 

dS  f  dv 


[        dS         ^  ^  [  dy 

]a  +  bQOsd       "  ]a  ■\-  b  ■¥  {a  —  b)f' 


§  III.]    MISCELLANEOUS    TRIGONO^rETRIC  INTEGRALS.  4 1 

The  form  of  this  integral  depends  upon  the  relative  values 
of  a  and  b.  Assuming  a  to  be  positive,  if  b,  which  may  be 
either  positive  or  negative,  is  numerically  less  than  a,  we  may 
put 

a  +  b       o 

7  ='^- 

a  —  b 

The  integral  may  then  be  written  in  the  form 

dy 


a  —  b  ]  (^  -]-  y^^ 
the  value  of  which  is,  by  formula  [k'), 


y 

tan-^-. 


c  {a  —  b) 
Hence,  substituting  their  values  for  y  and  c,  we  have,  in  this 


case, 

I -, a~ — T-^ — 79xtan-M  A/'^ rtani^ 

]a  +  bcosd      ^{a^  —IP')  \_y  a  +  b  _ 


.    .  {G) 


If,  on  the  other  hand,  b  is  numerically  greater  than  a,  this 
expression  for  the  integral  involves  imaginary  quantities;  but 
putting 

b  +  a 


b  ~  a 
the  integral  becomes 


^, 


•  dy 

the  value  of  which  is,  by  formula  {A'\  Art.  17, 

I        ,      c  +y 
log- 


c{b~d)     ^c-y' 


4-  FIEMENTARY  AfETHODS  OF  INTEGRA  770X.     [Art.  35. 


L- 


Therefore,  in  this  case, 
dd  I  .       V{b  +  a)+ V{b-a)  tdLW^d 


log 


dcos6~  ViS^-  a^)     ^  V{d  +  a)- V{b-a)  tan  ^0 
36,  U  c  <  I,  formula  (G)  of  the  preceding  article  gives 

de  2 


.     .  {G'] 


f— ^ 

J  I  +  r 


cos  0       V  {I  —  i^) 


tan' 


j/rT7^^"^^J-  •  (') 


Puttiiifi 


^i f.tani^=tani<^, (2) 

and  noticing  that  ^  =  o  when  ft  =  o,  we  may  write 

dd  (f> 


COS  0      V {i  —  c^)'    ' 


■     .     .     (3) 


Now,  if  in  equation  (i)  we  put  ^  for  6  and  change  the  sign  of 
c,  we  obtain 


It-- 


d<t> 


tan 


'I  +  ^ 
e 


y  j^— ;  tan  \ 


\^\ 


C  cos  ^  \'\  I   —  i''^) 

hence,  by  equation  (2), 

Equations  (3)  and  (4)  are  equivalent  to 

dB         ^        d(!> 
I +^  cos  (9       |/(i-i^)' ^5) 

•         d(f)  dB 


and 


§  III.]  TRIGONOMETRIC  INTEGRA  IS.  43 

the  product  of  which  gives 

(i  +  ^  cos  6)  {i  —  e  cos  (j))=  I  —  e^  .     .     .     .     (7) 
By  means  of  these  relations  any  expression  of  the  form 

f  dd 

J(i  +  ^  cos^)"' 

where  n  is  a  positive  integer,  may  be  reduced  to  an  integrable 
form.     For 


de 


dd 


J  ( I  +  £"  cos  8)"       J I  +  £"  cos  6^  ( I  +  ^  cos  6)"  - '  ' 
hence,  by  equations  (5)  and  (7), 

7 — , Zw  =  ; 2V7; — i     (i  —  ^  cos  6)"-^  d^ . 

J„  (i  +  ^  cos6^)«      (i/— ^)""Mo 

By  expanding  {i  —  e  cos  ^)""',  the  last  expression  is  reduced 
to  a  series  of  integrals  involving  powers  of  cos  (f> ;  these  may 
be  evaluated  by  the  methods  given  in  this  section  and  Section 
VI,  and  the  results  expressed  in  terms  of  6  by  means  of  equa- 
tion (2)  or  of  equation  (7). 

Examples  III. 

4         ,  tan'  mx      tan  mx 

1 .  tan  mx  ax,  — h  x. 

J  ^m  m 

2.  IdLXi  xdx,  A  ~  i  log  2. 

f      4  /c   ,      \  ^.  tan'(o  +  «')  ,,         , 

3.  I  sec*  (9  +  a)  d%  5^ '-  +  tan  (0  +  a). 


44  ELEMENTARY  METHODS  OF  IXTEGKA  TIOX.    [Ex.  III. 

4.        sin'  fnx  dx,  3^ 

Jo 


sm  6      sin  0 
3  5 


2     .    a^       4    .  z  ,    ,     2     .  ii. 

-  sin^Q sin*  en sin  « 

.^  7  XI 


2 

35 


■|  cos"  0  —  2   COS-  0. 


5.  sin' 0  cos' G  di'O, 

6.  L'(sin  e)  cos*  S  </0, 

IT 

7.  COS*  fl  sin'  G  dTs, 

f  sin'  0  </o 
J  V (cos  0) ' 

9.       7-^ i — ,        Multiply  by  sin'  0  +  cos'  d.        tan  0  —  cot  0. 

^    J  sin'  0  cos  0  -^  -^   -^ 

f  sin'  .V  c      /f  /     Q  t^"*  -^' 

10.  — r— ax.  See  A rl.  26.  . 

J  cos  -v  4 

11.  \^—. — - — ;— ,  i  (tan' Q  —  cot' &)  +  2  loe  tan  0. 
J  sin'  0  cos'  G  "  /  & 

(i/{s\nO)dO  3 

12.      i ,  |tan*0. 

J       cos*  0  ^ 

f  sin'jr  dx 

13-      r. , 

J    cos  X 

f  sin'x  </j:  tan'  x      tan'  ^ 

14-      f ,  + . 

J    cos  j;  5  3 

15.      sin' 6  cos' 0  ^0,  ^  [2O  —  sin  29C0S  20]. 


5  cos  X      3  cos  X 


III.] 


EXAMPLES. 


45 


1 6.    I     sin  7nx  dx, 

o 

sin^  S  dQ 


Tt 

2m 


17 


•1 


i8. 


19. 


cosO 

^      sine 
3 

dQ 


21. 


sm  0  +  cos  9 

dx 
I  +  cos  x^ 

dx 
Zi  —  cos  x^ 

dx 


log  tan    — 1 —     —  sm  i9. 
L4       2j 

i(log3  -i). 


tan  ^x. 
I  —  cot  \x. 


I  ±  sm  .r 
Miiltihly  both  terms  of  the  fraction  ^  i  T  sin  x.  tan  x  ±  sec  :j;. 


^/'i 


sec  0  ±  tan  Q 


log  tan  1^^ 


+ 


±  log  cos  &. 


24.      cos  0  cos  30  ^9.     iSi?^  Art.  33.  ^  sin  40  +  ^  sin  2^. 


25.  cos  6  cos  20  ^9, 

•^  o 

TT 

26.  sin"  0  sin  29  ^9, 

Jo 

TT 

27.  I    sin  3O  sin  29  d% 


\  sin*  9 


46  ELEMENTARY  METHODS  OF  INTEGRATION.     [Ex.  III. 


28.  sin  wO  cos  «0  </0, 

^  o 

I  —  COS  (»/  +  «)  0       I  —  COS  (m  —  n)  0 

2  (w  +  «)  2  (/«  —  «) 

29.  COS  X  cos  2a:  cos  3JC  ^, 

Reduce  products  to  sums  by  meatis  of  equation  (2),  ^r/.  33. 

I  Fsin  dx      sin  a^x      sin  2jc         ~| 
4L    6  4  2  "^J" 

30.  >i/  {\   —  CC)%x)dx,  2^/2. 

f  ^^  1        if^        ~l 

31.  -5 z ,.,    ■  ., —  ,  -7-tan~M  -tan^   . 

'^      ]a  cos';c  +  b  sin  .r  a/?  L«  J 

{       dx  I  , tan  ^ 

J I  +  cos  x  ^2  ^2 

f dx^ I  a  -\-  b  tan  0 

^'''   J  a'  cos"  X  —  b''  sin'  :c '  2  fl!/J     ^  a  —  ^  tan  9  " 

fsin  .r  ^jf  1  < ,  1 

—77 5 ; r-^ — 7,  COS""    H  cos:rf. 

V  (3  cos  jc  +  4  sin'  jc) '  <  8  ) 


fsin  X  cos'  :r  </jtr 
J  I  +  a'  cos'  X  * 


Putting  y  for  cos  ;c,  M(?  integral  becomes  — 


I  +  ay 


cos  ^       tan  ^  {a  cos  jv) 


§  III.] 


EXAMPLES. 


47 


^'■\-c 


dB 


Put  sin  0  =  cos  (5  —  |-7r),  and  use  formulas  {G)  a?id  (G). 


^f^>^'   Via^-b^) '''''" 


/a  —  b         2O  —  Tt 

tan 

4      J 


_  r'  a  +  b 


Ua<b, 


I  V  (^  +  g)  +  V  (^  -  ^)  tan  (p  -  i  7t) 

V  {b'  -  a')   ^^  V{b  +  a)  -  V  {b  —  a)  tan  (|  0  -  i  tt)" 


37- 


38. 


39- 


40. 


^9 


3 

+ 

5  cos  0  ' 

de 

5 

+ 

3  cos  0  ' 
^0 

5 

4  cos  9  ' 
^0      , 

ilog 


2  +  tan  i  0 


2  cos  9 


41 


■f 


^9 


42. 


3  —  cos  G 

dQ 


2  —  tan  i  0 
■I  tan"  ^[ I  tan  io]. 

f  tan" ^{3  tan|  0|. 

-1-  log  I  -  4/3  tan  jo 

V3      ^  I  +  •  4/3  tan  I-  6  ' 

tan"^  4/2 


2  ^3" 


4'?.      7 rr, ,     See  Art.  36. 

^^     J(i  +  ^cosO)-'  "^ 


cos 


J  ^  +  cosO 


sin  6 


^*-  II  u 


de 


(i  _  ^'J'^t  I  +  ^  cos  0       I  —  e    1  +  e  cos  9 

(2  +  /)  ;r 


(i  +  ^cos9)" 


2  (i  -  e')^ 


4S  ELEMENTARY  METHODS  OF  INTEGRATION.  [Ex.  III- 


4  S  • T-^ —  dXy 

•^     J  acosx  +  osmx 


Solution : — 

By  adding  and  subtracting  an  undetermined  constant,  the  fraction 
may  be  written  in  the  form 

/  cos  X  ■¥  q  s\r\x  -\-  A  {a  cos  x  ■¥  bsinx) 

a  cos  a:  +  ^  sin  :x;  ' 

we  may  now  assume 

/  cos  X  -T  </  sin  X  +  A  {a  cos  x  +  b  sin  x)  =  k  {l>  cos  x  —  asm  x); 

the  expression  is  then  readily  integrated,  and  A  and  k  so  determined 
as  to  make  the  equation  last  written  an  identity.     The  result  is 


1 


f>  cos  .r  +  ^  sm  a:   -         ap  +  bq          bp  —  aq .      ,  ,    .       . 
,   .       iix  =     ,   ,    ,..  X  +  -'. 7.J  log  (a  cos  X  -\-  bsxxi  x). 


46 


.      — — -f- ,     See  Ex.  \^. 

]  a  -V  b  XdiXi.  X  ^ 


ax  b       ,       ,  ,    .       \ 

~i  +  "IT"; — 7^  *og  (a  cos  a:  +  ^  sm  x). 


a'  +  <J-'      a"  +  b' 

47.  Find  the  area  of  the  ellipse 

X  ^  a  cos  ^  _>»  =  ^  sin  <J. 


—  Otob     sin*  ^  d^^  =  TTiz^. 


48.  Find  the  area  of  the  cycloid 

X  =  a  (?/•  —  sin  ?/)  y  =  a  {\  —  cos ^O- 


a"      (i  —  cos  ipy  dtp  =  sa'Ti. 

J  o 


§111.] 


EXAMPLES. 


49 


49.  Find  the  area  of  the  trochoid       {b  <  a) 

X  =  a'/:  —  dsini/j  y  =  a  —  d  cos  ip. 

{2a'  +  b')  7t. 

50.  Find  the  area  of  the  loop,  and  also  the  area  between  the  curve 
and  the  asymptote,  in  the  case  of  the  strophoid  whose  polar  equation  is 

;'  =  «  (sec  0  ±  tan  6). 
Solution  : — 
Using  0  as  an  auxiliary  variable,  we  have 


x—a{\  ±  sino) 


y  ■=  a 


,    sinVj-l 

tanr^  ± , 

cos  oj 


the  upper  sign  corresponding  to  the  infinite  branch,  and  the  lower  to 
the  loop.     Hence,  for  the  half  areas  we  obtain 


+  d  I    sin  fjdf}  +  «■  I    sin"  (j  do  =  a" 


I  + 


;] 


and 


—a^      sin  0  dS  +  d'       sin^  S  </0  =  «"  |  i • 


50  METHODS  OF  INTEGRATION.  [Art.  n. 

CHAPTER    II. 
Methods  of  Integration — Continued. 


IV. 

Integration  by  Change  of  Independent  Variable. 

37.  If  ^  is  the  independent  variable  used  in  expressing  an 
integral,  and  y  is  any  function  of  x,  the  integral  may  be  ex- 
pressed in  terms  of  j',  by  substituting  for  x  and  dx  their  values 
in  terms  of  y  and  dy.  By  properly  assuming  the  function  j', 
the  integral  may  frequently  be  made  to  take  a  directly  integra- 
ble  form.     For  example,  the  integral 

f     X  dx 


3{ax  +  bf 
will  obviously  be  simplified  by  assuming 

y  =  ax  +  d 
for  the  new  independent  variable.     This  assumption  gives 

X  = ,  whence  dx  =—: 

a  a 

substituting,  we  have 


f     X  dx       _  I    r(  J  —  b)  dy 
J  {ax  +  bf  ~  d'  J        'f 


I    ,  b 


§  IV.]  CHANGE   OF  INDEPENDENT    VARIABLE.  5 1 

or  replacing  jj/  by  x  in  the  result, 

\     X  dx  \  .       .  J.  b 

J-(5FT^  =  ^ '°g  ("^  +  *)  +  ?O^FT*)  • 

38.  Again,  if  in  the  integral 

f    dx 


j£^  —  I 

we  put  y  =  £%     whence 

X  =  log  y,  and  dx  =  — 

°  "^  y 

we  have 

dy 


Jf"^—  I      J 


Hence,  by  formula  {A\  Art.  17, 

I       ■^-  =  log-^^-^^=:  log  (£"— i)  —  ;tr. 


]e  —  \  y 

It  is  easily  seen  that,  by  this  change  of  independent  variable, 
any  integral  in  which  the  coefificient  of  dx  is  a  rational  func- 
tion of  £',  may  be  transformed  into  one  in  which  the  coefificient 
of  ^  is  a  rational  function  of  j. 


Transformation  of  Trigonometric  Forms. 

39.  When  in  a  trigonometric  integral  the  coefificient  of  dB  is 
a  rational  function  of  tan  ^,  the  integral  will  take  a  rational 
algebraic  form  if  we  put 

dx 
tan  6  =  X,  whence  dd  =  — - — s-  • 

'  I  +;r 


52  METHODS  OF  INTEGKATIOX.  [Art.  39. 

For  example,  by  this  transformation,  we  have 

f       d(^        _  f  dx 

J I  -^  tan^y"  J  (I  +.i^)(i  -Vx)' 

Decomposing  the  fraction  in  the  latter  integral,  we  have 

f        '■^^        _  ^f    '^-^'      _  '  (^II^L_       U_J^£_ 
J I  +  tan  ^  ~  2J I  +  A-*       2  J I  +  .1  -        ili+x 

=  I  tan"\-  —  i^  log  (l  +  .t^)  +  ^  log(i  +  x) 

or         f ^— -  =  U^  +  log  (cos  ^  +  sin  ^i)]. 

J  I  +  tan  ^  ^  ^  ^  ^-' 


40.  The  method  given  in  the  preceding  article  may  be  cm- 
ployed  when  the  coefficient  of  df^  is  7i  lioinoi:;cncons  rational  func- 
tion of  sin  /9  and  cos  B,  of  a  degree  indicated  by  an  even  integer  ; 
for  such  a  function  is  a  rational  function  of  tan  B.  It  may  also 
be  noticed  that,  when  the  coefficient  of  dd  is  any  rational  func- 
tion of  sin  B  and  cos  B,  the  integral  becomes  rational  and  alge- 
braic if  we  put 


s  =  tan 


for  this  gives 


sin  8  = . ,  cos  ^  = —2 ,  dd  = 


^' 


I  +  C  I  +  ^  I  +  ^ 

This  transformation  has   in  fact  been  already  employed  in 

the  integration  of .     See  Art.  35. 

a  +  b  cos  6  ^^ 


§  IV.]      LIMITS  OF   THE    TRANSFORMED  INTEGRAL. 


53 


Lmiits  of  the    Transformed  Integral. 

41.  When  a  definite  integral  is  transformed  by  a  change  of 
independent  variable,  it  is  necessary  to  make  a  corresponding 
change  in  the  limits.     If,  for  example,  in  the  integral 


we  put 


X  =  a  tan  6, 


[""       dx 

whence 


dx  —  a  sec^^  dB, 


we  must  at  the  same  time  replace  the  limits  a  and  co ,  which 
are  values  of  x,  by  \n  and^Tr,  the  corresponding  values  of  B. 
Thus 


"^       dx 


rr 


cos^  BdB 


2C?  _ 


+  sm  B  cos 


7t  —  2 


The   Reciprocal  of  x  taken   as    the   New  Independent 

Variable. 

42.  In  the  case  of  fractional  integrals,  it  is  sometimes  use- 
ful to  take  the  reciprocal  of  x  as  the  new  independent  variable. 
For  example,  let  the  given  integral  be 


dx 


Putting 


I 

;ir  =  -, 

y 


}x^{x+  if 
whence 


dj/ 


7 


,2  > 


54  METHODS  OF  INTEGRATION.  [Art.  42. 

wc  have 

Transforming  again   by  putting  z  =  y  -\-  \,  the  integral  be- 
comes 


{ds      {dz 


[{z-\f    ,  [      ,  f  ,  [dz       [ 

~  J       ^       fl'^  =  -  J  ^^rr  +  3  J  ^^  -  3  J  -  -f-  J 


=  -  7  +  3-  -  3  log  -  -  - 


Therefore,  since  z  =y  +  i  =  -  +  i  = 


X  +  I 
X 


\^{x^-\f~        '     2X'        '^~       X  X+I         ^^        X 


A  Power  of  X  taken  as  the  New  Independent  Variable, 
43.  The  transformation  of  an  integral  by  the  assumption, 

y  —  x"" (i) 

is  not  generally  useful,  since  the  substitution 


X  =  jj'",  whence  dx  —  -  y*"    ^  dy, 

n 

will  usually   introduce  radicals.     Exceptional  cases,  however, 


*  §  IV.]  THE  EMPLOYMENT  OF  POWERS  OF  X.  55 

occur.     For,  since  logarithmic  differentiation  of  equation  (i) 
gives 

dx      dy  ,  , 

—  =  — , (2) 

X      ny 

it  is  evident  that,  if  the  expression  to  be  integrated  is  the  product 

dx 
of  —  and  a  function  of  x^ ,  the  transformed  ex-^ression  will  be 

dv 
the  product  of  —  and  the  like  function  of  y. 

For  example,  the  expression 

{x^  —  i)  dx 

X  {x^  +   l)  ' 

dv 
which  is  the  product  of -^  and  a  rational  function  of  ;tr*,  becomes 

X 

dy, 


4yiy+  I) 


a  rational  function  of  y.     Hence,  decomposing  the  fraction  in 
the  latter  expression,  we  have 

{x^-i)dx^i  f   .r-i    ^.^Jio^  (y  +  'f 

x{x^+i)        4}y{y+i)''       4     ^         y 


=  loi 


V{-r*  +  i) 


X 


44.  When  this  method  is  applied  to  an  integral  whose  form 
at  the  same  time  suggests  the  employment  of  the  reciprocal, 
as  in  Art.  42,  we  may  at  once  assume  y  z^x''^.  Thus,  given 
the  integral 


r dx^ 

]^X^L2  + 


^^y 


56 


METHODS  OF  INTEGRATION. 


[Art.  44. 


putting 

y  =  ;^^~^                whence 

<^.i-  _       dy 

we  obtain 

'  1 

°       dx        _       \r  ydy 
x*{2  -\-x^)~       3J.  2J+  I 

y      log  {2y  +  l)" 
-      6"^           12        _ 

°  _  2  -  log  3 

12 

45.  The  same  mode  of  transforming  may  be  employed  to 

dv 
simplify  the  coefificient  of  —  ,  when  this  coefficient  is   not  a 
^  X 

rational  function  of  .r".     Thus,  the  integral 

f        dx 

ixV{x^-^) 

will  take  the  form  of  the  fundamental  integral  (/ '),  if  we  put 


x'=f, 


whence 


dx       2  dy 


Making  the  substitutions,  we  have 


dx  _2f  dy  _    2         -I  y   _    2         -i fx 


Examples  IV. 


log  (2  +  .r)  + 


2  +  A- 


§IV.] 


EXAMPLES. 


57 


X  dx 


(i-.r)- 


2X  —   I 

2(1  -.v) 


f  x"  -  .V  +  I 


2X  +  I  log  { 2a-  +  I ) 


8(2.1+  i) 


°       .r^  dx 


■X  ^v  +  2)  = 


log;/  + 


4y-  2 


l0g2--. 


dx 


I  +  f -^ 


:tr—  (log  I  +  f-'^). 


6. 


</a.' 


£-''  —  £- 


1  ,       £-^  —  I 

-  log 

2  ^  £-»-  +   I 


£-^+  I  ' 


I  —  I02  2. 


1 


g-'^+   I 
I  —  «-- 


dx, 


f-^  +  2  log  (£•*  —  l). 


■  2  +  tan  0 


3  —  tan  9 
J  tan'  e  —  I ' 


^9, 


e  —  log  (3  COS  9  —  sin  e) 


I  ,       tan  9—1       9 

—  log ; 

4         tan  9  +  1       2 


II. 


tan' 9 


tan''  9  —  I ' 


I  ,      tan  e  —  I    ,    6 

-  log +  - . 

4     ^  tan  9  +  I       2 


12. 


cos  9  </9 


«cos  9  —  <5  sin  9' 


aB  —  b  log  (a  cos  6  —  b  sin  e) 

a'  +  b' 


58 


METHODS  OF  INTEGRATIOX.  [Ex-  IV. 


COS  0  </0 


'.     ; ,     Flit  (j  =  a  +  b. 

*    J  cos  (<*  +  &) 


(o  +  a)  cos  «  —  sin  « log  cos  (o  +  a). 


14. 


'  sin  (0  +  n-) 


^/O, 


sin  (0  +  fi) 

{(j  +  Z^)  cos  («  -  yS)  +  sin  {a  -  /3)  log  sin  (o  +  /?). 


tan  (0  +  a)  cos  0  di'O,    —  cos  0  +  sin  a  log  tan 


2O  +  2n'  +  ;r 


16. 


f"     cos  0  1/9 
\o  sin  («  +  &)' 


cos  a  log  (2  cos  a)  +  a  sin  o". 


17- 


3   cos  1  0   ,  I    1       4/2  +  2  sin  0 

^—  ^0,  — r  log  —. ^—' 

o    cos  0  y  2     °  V2  —  2  sin  0 


6  _log(3  +  2.^2) 

o  V2 


18. 


sin  ^Oe/e 


sin  6 


19. 


r     A-'  dx 


log  tan  - 


TT   +   0 


i  log  {a'  +  x')  + 


2  {a'  +  x') 


log 


\/(l    +   A-')  I 


AT 


7r 

4  Jo 


2.*' 


TT 


sin  2O  dO=  —r 
4  Jo  16 


I  —  log  2. 


IV.] 


EXAMPLES. 


59 


r     dx 

'    \x'{x+  I)' 
•   J  (I  -xyx' 


.25 


•1: 


dx 


26. 


27. 


28. 


29. 


h 


jx 


x'  {x'  +  2)  ' 
dx 


(a  +  ^.v^) ' 
dx 


(a-'  +  i)  ^x 


1,1         ,        X  4-  I 
— J  H log  

2X  X  °         X 


+  -; +  log 


2(1  —  xf       1  —  X 


I  —  A- 


—  ^    +  il0g(2/   +   l) 

4 


°  _  2  -  log  3 


I     ,  X 

log 


4a     "=  ^  +  bx' ' 


I  % 


«^«     *.r'*  +  ^«' 


tlog(A-^-i)-logA-. 


V. 

Integrals  Containing  Radicals. 

46.  An  integral  containing  a  single  radical,  in  which  the 
expression  under  the  radical  sign  is  of  the  first  degree,  is 
rationalized,  that  is,  transformed  into  a  rational  integral,  by 
taking  the  radical  as  the  value  of  the  new  independent  vari- 
able.    Thus,  given  the  integral 

f  dx 

J  I  +   V(;ir+  i)' 


6o 


METHODS  OF  INTEGRATION. 


[Art.  46. 


putting 

y=   |/(,r+   1), 

whence 

X  ^ y  —  I,                    and                    dx  =  2y  dy, 

we  have 

f          dx 

J  I  +  '^{x  + 

T)='\ 

'  ydy  _  ^ 

I  +y      "J 

</7  —  2 

r  dy 

I  +7 

=  2y-2  log(i  +y) 

=  21 

/(^r  +  I) 

-  2  log  [I  +V{x  +  I)]. 

47.  The  same  method  evidently  applies  whenever  all  the 
radicals  which  occur  in  the  integral  are  powers  of  a  single 
radical,  in  which  the  expression  under  the  radical  sign  is  linear. 
Thus,  in  the  integral 


1 


dx 


,(,v_i)5  +  (A'-i)i 


the  radicals  are  powers  of  {x  —  i)^ ;  hence  we  put  y  =  {x  —  i)^. 
and  obtain 

J,U--i)U(.r-i)'  Jo/+? 

=  6f'0'-  i)dv  +  6[  ^^^—z^  -3+61og2. 

Jo  '  Jo  J   +    I 

48.  An  integral  in  which  a  binomial  expression  occurs 
under  the  radical  sign  can  sometimes  be  reduced  to  the  form 
considered  above  by  the  method  of  Art.  43.  For  example, 
since 

f        dx 
)x{x^  +  l)i 


§v.] 


INTEGRALS  CONTAINING  RADICALS. 


6i 


fulfils  the  condition  given  in  Art.  43,  when  71  =  3,  the  quantity 
under  the  radical  sign  may  be  reduced  to  the  first  degree. 
Hence,  in  accordance  with  Art.  46,  we  may  take  the  radical  as 
the  value  of  the  new  independent  variable.     Thus,  putting 


whence 
we  have 


31^=.  ^  —  I,  and 


dx 


^4 


s^  dz 


—  I 


dx        42^  dz 


X  -3(^-1)' 


Decomposing  the  fraction  in  the  latter  integral  as  in  Art.  20, 
we  have  finally 

f         dx            2  ^     -if/  <!  ,     x*~l   ,    I  1      (;i^+  i)^—  I 
-r  =  -  tan   '    (-ir^  +  i)      +  ~  log  ^ \   .      . 


Radicals  of  the  Form  V{<^^  +  b). 

49.  It  is  evident  that  the  method  given  in  the  preceding 
article  is  applicable  to  all  integrals  of  the  general  form 

\x'^"^^'{ax^  +  by+'^dx, (I) 

in  which  m  and  n  are  positive  or  negative  integers.     These 
integrals  are  therefore  rationalized  by  putting 


y  =  \/{a:^  +  b). 


62  METHODS  OF  INTEGRATION.  [Art.  49. 


Putting  VI  =  O,  the  form  (i)  includes  the  directly  integrable 

50.  As  an  illustration  let  us  take  the  integral 
f  dx 


dx  _    y  dy 
X  ~ y^  —  d^' 


putting  y  =  V{^  +  «^), 

whence  x^  =  }^  —  a^,  and 

we  have 

Hence,  by  equation  {A')  Art.  17, 


dx  _  I   .      J  —  '^  _   I  1       ^(-1"^  +  rt^)  —  rt 

X  V{x^  +  cF)  ~2a^^  y  +  a  ~  ~2a  °^  4^(-t^  +  a^)  +  a ' 


Rationalizing  the  denominator  of   the  fraction  in  this  result, 
we  have 

V{x^  +  i^)-a_[\(x^  +  d^)-af 
Vix"  +  a')  +a~  x" 

Therefore 

\— J^^^-L  log  ^^i^l^tl^  .    .    .    .{N) 


§  v.]  INTEGRALS  CONTAIiVIiVG  RADICALS.  63 

In  a  similar  manner  we  may  prove  that 

in 


J  X  V{a^  —  x^)       a 
61.  Integrals  of  the  form 


dx  \  .      a  —  V(a^  —  x^) 

=  -log ^ \     .     .     . 


[x'^"'{ax^^  by  +  idx (2) 


are  reducible  to  the  form  (i)  Art.  49,  by  first  putting  j  =  -. 

For  example : 

e       dx 


{a^  +  bf 


is  of  the  form  (2) ;  but,  putting  x  =  -  ,  whence 

y 


^(^^  +  ^)  ^  lllL+i^)  and  dx^-%, 

y  r 


we  obtain 


dx        _       [      y  dy 

7^* 


f         ^^         —  _  f 


{ax"  +  bf  J  {a  +  bfy^ 


The  resulting  expression  is  in  this  case  directly  integrable. 
Thus 

\       ^'^       —  ^  —  ^  [^\ 

iiax'  +  d)i~^V{a  +  bf)      bViax'  +  b)'     '     '    ^-^  ^ 


64  METHODS  OF  INTEGRATION.  [Art.  52. 

clx 

Integ^7'atio7i  of  — — -^ ^ . 

^  •'   V(^  ±  ^  ) 

52.  If  we  assume  a  new  variable  ^  connected  with  x  by  the 
relation 

z-x^  f(.i^±  A (0 

we  have,  by  squaring, 

:?  —  2ZX  =  ±  rt^, (2) 

and,  by  differentiating  this  equation, 

2  (^  —  .1)  dz  —  2z  dx  =  o ; 
whence 

dx    _  dz 

Z  —  X        z  * 

or  by  equation  (i), 


dx  dz 


(3) 


Vix"  ±(^)       z 

Integrating  equation  (3),  we  obtain 

63.  Since  the  value  of  x  in  terms  of  z,  derived  from  equa- 
tion (2)  of  the  preceding  article,  is  rational,  it  is  obvious  that 
this  transformation  may  be  employed  to  rationalize  any  ex- 
pression   which    consists   of   the  product  of— tt-s 5.    and  a 

^  V{^  ±  or) 

rational  function  of  x.     For  example,  let  us  find  the  value  of 
^Vi^±a')dx, 


J  4  J 


ds 


a^  {dz      a^  [dz 


4J  2   l  z         4  J 


4  J^ 


^  —  rt^        <?-  , 


§  v.]  TRIGONOMETRIC   TRANSFORMATION.  65 

which  may  be  written  in  the  form 

By  equation  (2) 

whence 

^^±^=^-^^)' (5) 

Therefore,  by  equations  (3)  and  (5), 

(^2  ±  a'f 


By  equations  (4)  and  (5),  the  first  term  of  the  last  member 
is  equal  to  \  x  s/{x^  ±  «-).     Hence 

s/^x'  ±  a')  dx  =  "^  ^(^^.  ^'),  -t-  ^'  log  Ix  +  V{x^  ±  (f)]  .    .    (Z) 


Transformation  to   Trigonometric  Forms. 

64.  Integrals  involving  either  of  the  radicals 
V{a^~x^),  Via^  +  x"),  or  Vi^"  -  c^) 


66  METHODS  OF  INTEGRATIOh\  [Art.  54.. 

can  be  transformed  into  rational  trigonometric  integrals.     The 
transformation  is  effected  in  the  first  case  by  putting 

X  —  a  ^\r\.  B,  whence  V(<z^  —  x^  =  a  cos  6 ; 

in  the  second  case,  by  putting 

.r  =  rt  tan  B,  whence  V(<r^  +  x^)  —  a  sec  B  ; 

and  in  the  third  case,  by  putting 

X  =  a  sec  B,  whence  \'{x^  —  (i^)  =  a  tan  B. 

55.  As  an  illustration,  let  us  take  the  integral 

Via"  -  x^)  dx  ; 

putting  ^  =  rt  sin  B,  we  have  -/(^^  —  x^)  =  a  cos  B,  dx  =  a  cos  B  dB ; 
hence 


I  ^(^  _  x^)  dx  =  a'' 


cos^  BdB 


a-  B     (^  sin  B  cos  B 

=  T+ 2 ' 

by  formula  {C)  Art.  29.     Replacing  B  by  x  in  the  result, 

\  I  I  ji        2.\  J         c^    .       XX  V(d^  —  x^  ,  ,  ^  X 

\  V  (a'  —  xr)dx  ~  —  sin  - '  -  h .     .     .     (J/) 

J  2  a  2 

Regarding  the    radical    as   a  positive   quantity,  the  value 
of    B  may  be   restricted  to  the  primary  value  of  the  symbol 

sin  - '  -  (see  Diff.  Calc,  Art.  54) ;  that  is,  as  x  passes  from  —  a 

to  +  a,  B  passes  from  —  J  ;r  to  +  ^7t. 


§V.] 


INTEGRALS  CONTAINING  RADICALS. 


67 


Radicals  of  the  Form  '\/(a:^  +  bx  +  c). 

56.  When  a  radical  of  the  form  ^/(^.r^  +  dx  +  c)  occurs  in  an 
integral,  a  simple  change  of  independent  variable  will  cause  the 
radical  to  assume  one  of  the  forms  considered  in  the  preceding 
articles.     Thus,  if  the  coefificient  of  x^  is  positive, 


V{ax^+  dx  +  c)  =  Vay 


2a)  40^ 


in  which,  if  we  put  ;i' +  —  =  y,   the   radical   takes   the    form 

2a 

V{y^  +  a^)  or  V{j"  —  a^),  according  as  4ac  —  d'^  is  positive  or 
negative.  If  a  is  negative,  the  radical  can  in  like  manner  be 
reduced  to  the  form  \/(a^  —  J^)  or  V(—  a^— y^) ;  but  the  latter  will 
never  occur,  since  it  is  imaginary  for  all  values  of  y,  and  there- 
fore imaginary  for  all  values  of  x. 

For  example,  by  this  transformation,  the  integral 


dx 


{ax^  +  bx  +  c-)'3 


can  be  reduced  at  once  to  the  form  (y),  Art.  51.     Thus 
dx  r  dx 


\{ax^ 


+  bx-\-  cf^ 


b\-       Aac  —  b^ 

a[x  +  —)    +  ^ 

2a/  4a 


x  + 


2a 


4ac 


U" 


4a 


'f/(ax^  +  bx  +  c) 


4ax  +  2b 

{4ac  -  l^)  Viax^  +  bx  +  c) 


68  METHODS  OF  IXTEGRATIOX.  [Art.  57. 

57.  When  the  form  of  the  integral  suggests  a  further 
change  of  independent  variable,  we  may  at  once  assume  the 
expression  for  the  new  variable  in  the  required  form.  For 
t;xample,  given  the  integral 


I  ^{2a.\ 


t^)  X  dx ; 

we  have  V(2ax  —  .1^)  =  V[a^  —  {x  —  af] 

hence  (see  Art.  54),  if  we  put  x  —  a  =  a  sin  $,  we  have 
\\2ax  —  x^)  —  a  cos  6, 
X  =  a{i  +  sin  fi),  dx  =  a  cos  8 dB  ; 


de 


\\  V{2ax  —  x-)xdx  =  «^    cos^  ^(i  +  sin  ^) 

=  —  (^  +  sin  ^  cos  ^)  -  —  cos^  6 

=  —  sin-'  ±J~f  +  ^(.r  -  a)  V(2ax  -  .v^)  -  -  (2ax  ~  .v^)» 
2  a  2  3 

^^        •  X  —   a  I        ,,  ox   r  « 

=  —  sm  -  • +  ^  V{2ax  —  x^)  \2.\P'  —  ax  —  3^]. 


1 


The  hitegrals 

dx  r  dx 

and 


58.  An  integral  of  the  form  I    ,.     ,  ^^, may  by  the 

method  of  Art.  56,  be  reduced  to  the  form  (K),  Art.  52,  or  to 
the  form  (/'),  Art.  10,  according  as  a  is  positive  or  negative. 


§  v.]  IRRA  TIONAL  INTEGRALS.  69 

But  when  the  quantity  under  the  radical  sign  can  be  resolved 
into  linear  factors,  the  formulas  deduced  below  give  the  value 
of  the  integral  in  forms  which  are  sometimes  more  convenient. 
If  a  and  (i  are  the  roots  of  the  equation 

ax^  +  bx  +  c  =  o, 
the  integral  may  be  put  in  the  form 


I 


dx  \        [  dx 

or 


V[(^'  -a){x-  p)\  V{-d)]  i/[(.r  -  a){ii  -  x)\   ' 

according  as  a  is  positive  or  negative.     Assuming 

\/{x  —  a)  —  z,       whence         x  =  ^  -k-  a         and         dx  =  2zdz, 

we  have 

by  formula  {K),  Art.  52  ;  hence 

1  V[(.  -  %(.  _;>)]  =  ^  log  [  ^(■'-  '^)  +  ^(^  -  ffl  ■     •    -W 

In  like  manner  we  have 

f dx^ f  dz  __       .  _^         z 

J  VV{x-a){(i  -x)-\-^  ]^{fi-a-z')  -  ^  ^'''~    V{P  -  a)  ' 

by  formula  (/')  ;  hence 

f  dx  .     ,  ./  X  —  a  ,^. 


/O  METHODS  OF  INTEGRATION.  [Art.  58. 

It  can  be  shown  that  the  values  given  in  formulas  (A^)  and 
{O)  differ  only  by  constants  from  the  results  derived  by  em- 
ploying the  process  given  in  Art.  56. 


Examples  V. 

s/{a  —  x)-x  dx,  {a  —  x)^  (3.V  +  2a). 

:  jv{-r  +a)-x-'dx,  ^  (a  +  x)^ -^  {a  +  x)i  + —(a  +  x)l 

{    Xdx  2      3                                          ,        , 

•  Jm^'  -X^-X+2^X-2\0g{l    +YX). 

[xdx  2  /              ,    ,,           . 

•  J^Tf^T^'  -^{x-2a)V{x-^a). 

■  \^x-i'  ^  ^"'"  +  2  log  (i  -  Vx). 


6.    f    {a  +  x)Krdx,  ^-^^Y=.-9A. 

J-a                      '                  '  7                   4     Jo                     28 

[           dx  2               /2x  —  a 

7-      "t;? r\>  -tan  '4/ ■■ 

]x  \  \2ax  —  a  )  a             f         a 

8.    \\a-x)^^x^dx,  _^:l^^_al__2dyy     ^16^, 

Jo  9           7           5    JV'^       315 

J  2x^  —  x^  44                  8 


§  v.]  EXAMPLES.  yi 

Jo  o         5  Ji  lo        40 

''•   1^(7-% ^-^^  2  (1+  :r)  V{i  -  x). 


Rationalize  the  denominator. 


3«' 


2  (jc  +  a)'2  —  2  {x  +  iJ)2 


t/(^v  +  tz)  +  4/(^  +  ^) '  3  (a  -  ^) 

I,       V(^'  +  i)  -  I 


''■[-^v^^^y  i^"^ 


4     ^  4/(a;*  +  i)  +  i' 
I  i/(-r^  +  i)  dx  V{x'  ^1)    ,    i,_  V(a-^+  i)-i 

'5-  I        -        '  2       ■*"4'°^v(a-^  +  i)  +  r 


16. 


(a'^+i)  (.r'^-  i)t 


^x, 


nV-        $  3  J 

f'^      ^V.r  /  ,         a'"]''^"       rS       11^2-1    , 


4/(a:''  —  a")  —  «  sec  - '  -  . 


72  METHODS  OF  INTEGRATION.  [Ex.  V. 


f       x^dx 
■  ]>/{x'^ay 

—  .   3 ^dx.     See  formulas  (L)  and  {K). 


v'  a-  xt"  _  /T*  .  "1 


-xVix'  +  a')  -  -  ^Mog  [.V  +  ^{x'  +  a')]  . 

2  2 


5.    I  -^^—^. dx,  a  log ^- +  Via'  -  x  ) 

f  dx 

'■  ]x  +  vix'  ^-dy 

^  Vix^  +  a^)  +  '-log  [.r  +  Vix^  +  a')]  -  ^ 

log  [  Vix'  +  a')  +  x]-  ^^^ '- 

f  dx 

^-  ]Vix'  +  0")  -a  ' 


log  [  Vix'  +  fl')  +  ^J  -  ^  ^    - 


or  x 


24-    f-77^T^-     -S"^^  Formula  {K).      -log  [a:'  +  V(i  +  ^-*)]. 


§v.j 


EXAMTLES. 


73 


25.      /^{ax^  +  b)  dx,  \a  >o]         Fui  Viax"  +  b)  =  z  —x  Va. 

— -log  LxVa  +  Viax^  +  ^)]  +  -  xViax'  +b). 
2  ya  2 


26. 


J  {a  +  a-)  \/{x' 


+  b-')' 


I j^   ^  +  V{x'  +b')  +  a  -  Via'  +  b') 


27 


V[a'+b')  ^.r  +  Vi^r'  +  />')  +  a  +  V{a'  +  b') 
dx  V(i  +  x^) 


—  cot^ 


=  V3. 


29. 


x^  dx 


(-v=-«*)t' 


30-   J 


</je 


{p  +  qx)V(.x'  +  i)' 


i/(.v^  -  a') 


-77 — 5 rr  loe  tan  — 

4/(/  +  /)    ^         2 


tan-'  X  +  tan  ' 


31- 


V(:c'-i)' 


^(.t'  -  l)  _^  I 

■^^ ^ — ^  H —  sec  - ' ;»;. 

2.T  2 


32. 


«      .r"  </.r 


r- 

Jo  {X 


+  a')« 


log  tan  5^-^. 


74 


METHODS  OF  INTEGRATION. 


[Ex.  V. 


fl'.V 


.vV(-v'-i)' 
f dx 

^^'  J  (I  +A-')i/(i-a-")' 

^  ■  Jo  -/(a-Ji)  L     Jo  y(aA--a--')J 


(2A-^   +    l) 


3^' 


I  .         X^2 

—7-  tan-'  — t; r\  • 


a  X 


sees. 


V(a-^  -  i) 


4 


f  dx 

38.     f-1-^^ T  •  ^''^  ■^■'  =  2  = 

•5       Ja-V(.v*-i) 

39-   J^   V{2ax  -  x')-dx, 

40.  4/(2rt.v  —  A'')-.va'A-, 

«'      ^  cos'  0  (i  +  sin  0)  a'O  =  a' . 

2 

41.  'V/(2rt.V  —  A-')..!-"  </.V, 

a*|\cos=0(i +sin(/)Vo=a'    ^f  " -]  • 


v.] 


EXAMPLES. 


75 


42. 


dx 


43- 


44. 


J  '\/{2ax  +  x')  ' 

4y  ^r/.  56,         log  \x^-  a  +  V{2ax  +  .r^)]  +  C ; 

by  Art.  58,  log  [  Vx  +  V{2a  +  x)]  +  C\ 

f         X  dx 


^    2a  —  X        \_      J  4/(2a;i;  —  .v  )_ 


«z  sin-'  " V{2ax  —  .v*). 


f  Jy.  .      


^^y  ^/-A  58,  2  sin 


-,/ 


r  +  I 


+  C'. 


46 


■1: 


</;tr 


47 


f ^'^-^  ,  ^j^- 1  -^'-i  _  (-^-  +  3)  V(3  +  2a-  -  x') 

•  ]  S'{Z  +  2x-xy^''^^         2  ^  • 


2  sin' 


./g;- 


fza 
J  a 


^P 

V{2  —X  —  X')^ 


7t 
2 


</^ 


^{x"  —  ax) ' 


log  (3  +  2  V2). 


/O  METHODS  OF  INTEGRATION.  [Ex.  V. 

50.  Find  the  area  included  by  the  rectangular  hyperbola 

y  =  zax  +  -v', 
and  the  double  ordinate  of  the  point  for  which  x  =  2a. 

(i'\p\'2  -  log  (3  +  2  4'2)]. 

51.  Find  the  area  included  between  the  cissoid 

•V  (-v»  4-  /)  =  2ay- 

and  the  coordinates  of  the  point  {a,  a)  ;  also  the  whole  area  between 
the  curve  and  its  asymptote. 


f  —  TT  —  2  ja',        and         jTra' 


52.  Find  the  area  of  the  loop  of  the  strophoid 

x{x^+f)  +^(.v^-y)  =  o; 
also  the  area  between  the  curve  and  its  asymptote. 


2<2'  (  I  —  —  )  ,       and        2it 


(■  ^ :) 


For  the  loop  put  )■  =  —  .v  --      r.-.  ,  since  x  is  nes'ative  between  the  limits 

V{a  —  x")  ■^ 

^  a  and  o. 

53.  Show  that  the  area  of  the  segment  of  an  ellipse  between  the 

minor  axis  and  any  double  ordinate  is  ab  sin'" — V  xy, 

a 


§  vi.] 


INTEGRATION  BY  PARTS. 


77 


VI. 

Integration  by  Parts. 

59.  Let  21  and  v  be  any  two  functions  of  x\  then  since 

d  {uv)  ■=  u  dv  -\-  V  dUf 

uv  =    lidv  +   V  du. 


whence 


\u  dv  =  tiv  —   V  dii (i) 


By  means  of  this  formula,  the  integration  of  an  expression 
of  the  form  iidv,  in  which  dv  is  the  difTerential  of  a  known 
function  v,  may  be  made  to  depend  upon  the  integration  of 
the  expression  v  du.     For  example,  if 


we  have 


u-=  zo's,~''x  and  dv  =  dxy 

dx 


du  =  — 


v{\  -  x^y 


hence,  by  equation  (i), 


cos " '  :\ 


x-dx  =  ;trcos-^;f  + 


xdx 


in  which  the  new  integral  is  directly  integrable ;  therefore 

QO's-'^x-dx  =  ;f  cos"';ir  —  4/(1  —  x^). 
The  employment  of  this  formula  is  called  integration  by  parts. 


78 


METHODS  OF  INTEGRATION: 


[Art.  60. 


Geometrical  Illustration. 


60.  The  formula  for  integrat'on  by  parts  may  be  geomet- 
rically illustrated  as  follows.  Assum- 
ing rectangular  axes,  let  the  curve  be 
constructed  in  which  the  abscissa  and 
ordinate  of  each  point  are  correspond- 
ing values  of  v  and  «,  and  let  this 
curve  cut  one  of  the  axes  in  B.  From 
any  point  P  of  this  curve  draw  PR 
and  PS,  perpendicular  to  the  axes. 
Now  the  area  PBOR  is  a  value  of  the 


Fig.  2. 


indefinite  integral    ?^  ^z^,   and    in    like 


manner  the  area  PBS  is  a  value  of 
and  we  have 


V  du ; 


Area  PBOR  =  Rectangle  PSOR  -  Area  PBS ; 


therefore 


\ii  dv  =  iiv 


V  du. 


Applicatiojis. 


61.  In  general  there  will  be  more  than  one  possible  method 
of  selecting  the  factors  u  and  dv.  The  latter  of  course  in- 
cludes the  factor  dx,  but  it  will  generally  be  advisable  to  in- 
clude in  it  any  other  factors  which  permit  the  direct  integra- 
tion of  dv.  After  selecting  the  factors,  it  will  be  found  con- 
venient at  once  to  write  the  product  u-v,  separating  the  factors 
by  a  period  ;  this  will  serve  as  a  guide  in  forming  the  product 


§  VI.]  INTEGRATION  BY  PARTS.  79 

V  du,  which  is  to  be  written  under  the  integral  sign.     Thus,  let 
the  given  integral  be 


x"-  logf  X  dx. 


Taking  x^  dx  as  the  value  of  dv,  since  we  can  integrate  this 
expression  directly,  we  have 


„  dx 


x'^  log  X  dx  =  log  X-  —  x^ ; 

=  —  x^  log'  X x^  dx 

3  ^  V 


=  ^(3log^-i). 

62.  The  form  of  the  new  integral  may  be  such  that  a 
second  application  of  the  formula  is  required  before  a  directly 
integrable  form  is  produced.  For  example,  let  the  given 
'.ntegral  be 

x^  cos  X  dx. 

In  this  case  we  take  cos  x  dx  ~  dv,  so  that  having  ,x^  —  u,  the 
new  integral  will  contain  a  lower  power  of  x:  thus 

x^  cos  X  dx  =  .r^-sin  x  —  2    x  sin  x  dx. 

Making  a  second  application  of  the  formula,  we  have 

\x^cosxdx  =  ;t;*sin;i^— 2    x{-  cos x)  +    cos  xdx 

=  .r'sin  X  +  2x  cos  ;f  —  2  sin  x. 


80  METHODS  OF  INTEGRATION.  [Art.  63. 


63.  The  method  of  integration  by  parts  is  sometimes 
employed  with  advantage,  even  when  the  new  integral  is  no 
simpler  than  the  given  one  ;  for,  in  the  process  of  successive 
applications  of  the  formula,  the  original  integral  may  be  repro- 
duced, as  in  the  following  example: 

i"'^  sin  («.v  +  ix)  dx 

—  cos  inx  +  ff)      ?«  f  ,„  ^       /       ,     \  J 

^ — —  '  ^ —  £"'-^  cos  vnx  +  a)  dx 

n  n  ] 


=  £' 


f '"-^  cos  {nx  4-  a)     m  ^^^  sin(;/.r  +  n:)  _  w^ 

n  1^  . 


n  n 


^'"^  sin  {iix  +  a)  dx, 


in  which  the  integral  in  the  second  member  is  identical  with 
the  given  integral ;  hence,  transposing  and  dividing, 

£"'-^  sin  {nx  +  a)  dx  =  —^ — —3  \in  sin  {ttx  -^  a)  —  n  cos  {nx  +  «)]. 


64.  In  some  cases  it  is  necessary  to  employ  some  other 
mode  of  transformation,  in  connection  with  the  method  of 
parts.     For  example,  given  the  integral 

[sec^^^^; 

taking  dv  —  sec'^  B  dd,  we  have 

fsec8^^^  =  sec^-tan(9-  fsec^tan^d^d?^;.     .     .     (l) 


§VI.] 


FORMULAS  OF  REDUCTION. 


If  tiow  we  apply  the  method  of  parts  to  the  new  integral,  by 
putting 

sec  B  tan  B  dO  ~  dv, 

the  original  integral  will  indeed  be  reproduced  in  the  second 
member ;  but  it  will  disappear  from  the  equation,  the  result 
being  an  identity.  If,  however,  in  equation  (i),  we  transform 
the  final  integral  by  means  of  the  equation  tan^  B  =  sec^^  —  i, 
we  have 


sec''  Bdd  :^  sec  ^  tan  (9  - 


sec^  B  dB  i- 


sec  Bdd. 


Transposing, 


2  I  sec^  B  dd  ~  -^,-7^  + 
cos  B 


dd 


cos 


9' 


hence,  by  formula  {F),  Art.  31, 


lsec«  Bdd  =  -^^^^^  +  '-  log  tan  [-  4-  ^]. 


sin  B        I 
2  cos-  B      2 


Formulas  of  Reduction. 


65.  It  frequently  happens  that  the  new  integral  introduced  ♦ 
by  applying  the  method  of  parts  differs  from  the  given  integral 
only  in  the  values  of  certain  constants.  If  these  constants  are 
expressed  algebraically,  the  formula  expressing  the  first  trans- 
formation is  adapted  to  the  successive  transformations  of  the 
nev/  integrals  introduced,  and  is  called  a  formula  of  reductio7i. 


82 


METHODS  OF  INTEGRA  TIOX. 


[Art.  65. 


For  example,  applying  the  method  of  parts  to  the  integral 


we  have 


f  A"'  e""^ dx  -X"-—  --  I -r" -^^''''dx ( I ) 

J  a       a  ] 


in  which  the  new  integral  is  of  the  same  form  as  the  given 
one,  the  exponent  of  x  being  decreased  by  unity.  Equation 
(i)  is  therefore  a  formula  of  reduction  for  this  function.  Sup- 
posing ;/  to  be  a  positive  integer,  we  shall  finally  arrive  at  the 

integral  f -^  ^.i-,  whose  value  is — -.  Thus,  by  successive  appli- 
cation of  equation  (i)  we  have 


X"  «"■*■  dx  = 


a   L 


^n 


n 

-  X' 

a 


+  (-  0"  -^ — i. — 


Reduction  of  Iszn'"  6  dd  and 


cos 


ede. 


66.  To  obtain  a  formula  of  reduction,  it  is  sometimes  neces- 
sary to  make  a  further  transformation  of  the  equation  obtained 
by  the  method  of  parts.     Thus,  for  the  integral 

sin"'  ddd, 

the  method  of  parts  gives 

[  sin"'  ddd=  sin"' - '  ^  (-  cos  ^)  +  {m  -  i)  f sin'« "»  d  cos*  6  dd. 


§  VL]      REDUCTION  OF   TRIGONOMETRIC  INTEGRALS.  83 


Substituting  in  the  latter  integral  i  —  sin^  B  for  cos^  Q, 
sin'"  Odd  =  —  sin'"-^  6^cos^ 


+  {m  —  i)    sin'"-^  6 dd  —  {in  —  i) 


sin'«  B  dB ; 


transposing  and  dividing,  we  have 


„  ,„  sm"'-'  «9cos  B       in  -  I 

sin"'  BdB  = \ 

in  in 


^dB,    .     .     .     (I) 


a  formula  of  reduction  in  which  the  exponent  of  sin  B  is  dimin- 
ished two  units.  By  successive  application  of  this  formula,  v/e 
have,  for  example : 


sin' BdB 


sm'*  B  cos  B      5 
6 +  6 


sin' BdB 


sm^(9cos  B      5  sm^^cos  B      S  3  (  ■  9  n  j/^ 

— ^ ■  +  ^  -  sm^  B  dB 

6  64  64J 


sin^  ^  cos  6*      5sin^^cos/9      5 •  3  sin  (9  cos  (9      5 •3' I 


6.4 


6-4'2  6-4-2 


67.  By  a  process  similar  to  that  employed  in  deriving 
equation  (i),  or  simply  by  putting  B  =  ^it  —  B'  in  that  equa. 
tion,  we  find 


^  „      cos'"-'  6' sin  B       in  —  i 
cos'"  BdB  = ■  + 


m 


cos'"-'' BdB,   .     .     (2) 


a  formula  of  reduction,  when  in  is  positive. 


84  METHODS  OF  integration:  [Art.  68. 

68.  It  should  be  noticed  that,  when  in  is  negative,  equation 
(i)  Art.  66  is  not  a  formula  of  reduction,  because  the  exponent 
in  the  new  integral  is  in  that  case  numerically  greater  than  the 
exponent  in  the  given  integral.  But,  if  we  now  regard  the 
integral  in  the  second  member  as  the  given  one,  the  equation 
is  readily  converted  into  a  formula  of  reduction.  Thus,  put- 
ting —  n  for  the  negative  exponent  i)i  —  2,  whence 

in  =  —  n  +  2, 

transposing  and  dividing,  equation  (i)  becomes 

f   de  cosf) 


n-2  f     dd 
^  n-  \]^"--e'   •     •     •     ^3) 


J  sin"  (9  (;/—  i)sin"-'  B 

Again,  putting  B  —  \ti  —  B'  in  this  equation,  we  obtain 

f    dB    _  sin  B  n-2  f     dB 

J  cos"  B  ~  (n  —  i)cos"-'B      n—  i  J  cos" -"(9     *     '     *     *     W 


Reduction  of 


sin"'B  cos"  B  dB. 


69.  If  we  put  dv  —  sin'"  B  cos  B  dB,  we  have 
cos"-'6^sin"'+»^ 


sin"'  ^cos"  BdB  = 


+ 


in  +  I 
n  —  \    ' 


in  +  I  . 


suV"+^  B  cos"-'  BdB;  .     .     .     (i) 


but,   if  in  the  same  integral  we  put  dv  =  cos"  ^  sin  BdB,  we 
have 

sin"'-'^cos''+'  B 


sin'"  B  cos"  B  dB  =  — 


n  +  I 


+  —, — -  [sin'"-'' Bcos"-^'' BdB.    ...    (2) 
n  +  I  J 


§  VI.J       REDUCTION  OF   TRIGONOMETRIC  INTEGRALS.  85 

■ * 

When  in  and  n  are  both  positive,  equation  (i)  is  not  a 
formula  of  reduction,  since  in  the  new  integral  the  exponent 
of  sin  B  is  increased,  while  that  of  cos  d  is  diminished.  We 
therefore  substitute  in  this  integral 


sin'"+^  B  =  sin"'  ^  (i  —  cos^  6), 
so  that  the  last  term  of  the  equation  becomes 


ji  —  I 
7n  +  I 


sm'"  a  cos" 


'de~' 


n  —  I 


in  +  I 


sin'«  6  cos^ddd. 


Hence,  by  this  transformation,  the  original  integral  is  repro- 
duced, and  equation  (i)  becomes 


fi  + -^      f  sin'«  6  cos"  Odd  ^ 

[_        m  +  I  j] 


m  -\-  I 


+ 


n  —  I 


sin"'  dcos"-'ddd. 


Dividme  by  i  H = ,  we  have 

^    "^  m  +  I       m  +  I 


sin'"  6  cos''  edd  = 


sin'^+'/^cos"-^! 
m  +  It 


+ 


n  —  I 
fn  +  n  , 


sin'"  6  cos"-^  0  d6, 


(3) 


a  formula  of   reduction    by  which  the  exponent  of  cos  6  is 
diminished  two  units. 


86  METHODS  OF  INTEGRATION.  [Art.  69. 

^ . 

By  a  similar  process,  from  equation  (2),  or  simply  by  put- 
ting d  —  \n  —  6'  m  equation  (3),  and  interchanging  in  and  «, 
we  obtain 

f    .       .,  r^   m  sin"' " ' /9  cos"+' ^ 

sin'"  d  cos"  edd— 

J  m  +  71 

m  -\-  n  ]  ^ 

a  formula  by  which  the  exponent  of  sin  B  is  diminished  two 
units. 

70.  When  n  is  positive  and  in  negative,  equation  (i)  of 
the  preceding  article  is  itself  a  formula  of  reduction,  for  both 
exponents  are  in  that  case  numerically  diminished.  Putting 
—  in  in  place  of  in,  the  equation  becomes 


cos"  B  ,„  cos""'  B  n  —  I 


^?^—dB.    ...    (5) 


fcos^^^^_ 

Jsin'"^  (in  —  1) sin"'-' B      in  —  i  . 

Similarly,  when  in  is  positive  and  n  negative,  equation  (2)  gives 

fsin"'^    ,^_        sin'«-'/9  m—i   {s'm"'-'B 

J  cos"  ^  («  — i)cos"-'^      n 


if!in:!:i^,/,.  ...  (6) 

I   J  cos" -^6*  ^  ^ 


71.  When  in  and  11  are  both  negative,  putting  —  in  and  —  n 
in  place  of  in  and  ;/,  equation  (3)  Art.  69  becomes 


I 


dB 


sin'"  B  cos"  B  {in  +  n)  sin'"- '  B  cos"-^'  B 

dS 


+ 


;/  +  I  f 
111  +  7/  J  si 


111  +  11  ]s'n\"'  Bcos"'^-B^ 
in  which  the  exponent  of  cos  B  is  numerically  increased.     We 


§  VI.]      REDUCTION  OF   TRIGONOMETRIC  INTEGRAIS. 


87 


may  therefore  regard  the  integral  in  the  second  member  as  the 
integral  to  be  reduced.  Thus,  putting  71  in  place  o{  n  -\-  2,  we 
derive 


dd 


I  sin"'  6  cos«  B      (;^  —  i )  sin'«  - '  i9  cos"  - '  ^ 


+ 


7n  +  n 


dd 


svs\"^  c/  cos'^ 


n  —  \ 
Putting  6  =  ^Tt  —  6',  and  interchanging  m  and  n,  we  have 

f        dd I 

J sin'«  9 cos«  d~      (m—i) sin"' -^6cos"-^d 

m  +  n  —  2  [  dd 


(7) 


+ 


vt 


n  —  2  ^ 
—  I      J  si 


sin'^-^l^cos"^ 


(8) 


72.  The  application  of  the  formulas  derived  in  the  preced- 
ing articles  to  definite  integrals  will  be  given  in  the  next  sec- 
tion. When  the  value  of  the  indefinite  integral  is  required,  it 
should  first  be  ascertained  whether  the  given  integral  belongs 
to  one  of  the  directly  integrable  cases  mentioned  in  Arts.  27 
and  28.  If  it  does  not,  the  formulas  of  reduction  must  be 
used,  and  if  ni  and  n  are  integers,  we  shall  finally  arrive  at  a 
directly  integrable  form. 

As  an  illustration,  let  us  take  the  integral 

f  sin^  e  cos-*  e  dd. 


Employing  formula  (4)  Art.  69,  by  which  the  exponent  of  sin  6 
is  diminished,  we  have 


sm"  {)  cos* 


sin  d  cos'  d      I 
6 +  6 


cos'»</«. 


88 


METHODS  OF  INTEGRATION. 


[Art.  72. 


The  last  integral  can  be  reduced  by  means  of  formula  (2)  Art. 
(i-j,  which,  when  ;;/  =  4,  gives 


4  cos"  ^sin  e      3 

cos*  6  d6  = 1 — 

4  4 


cos2  e  dS : 


therefore 

f    .  ,  ^       ,  ,,  „,      sin  6  cos^  d       cos^  ^  sin  (9  ,  sin  /9  cos  ^  ,    ^ 
Jsm=^cos'(^rf»  = g + i^— +  — 16—  +  16- 

73.  Again,  let  the  given  integral  be 
[  cos^  e  dd 

By  equation  (5),  Art.  70,  we  have 

•  cos«  Odd  _  _  Qos^  6  _  5^ f cos"  6  dd 
)    sin'  6    ~      2  s\v?  6      2  J     sin  ^    * 

We  cannot  apply  the  same  formula  to  the  new  integral,  since 
the  denominator  w—  i  vanishes  ;  but  putting  11  =-4.  and  m  —  —  i, 
in  equation  (3)  Art.  69,  we  have 


fcos"  Odd      cos"  0 


sin  6 


3 

cos"  0 
3 

cos"  8 


+ 


cos"  Odd 


sm 


+ 


Jsin  ^       J 


sin  Odd 


Hence 


+  log  tan  -  6  -¥  cos  6. 
3  '=2 


[cos^Bdft  cosM       5cos"^       5  ,      .      i  z.       5         .j 

I  — .  -  ^    = r-o:^  —  - — ^ log  tan  -  ^  -  -  cos  6. 

J     sm"  6  2  sm^  ^  6  2^22 


§VI.] 


EXTENSION  OF   THE  FORMULA. 


89 


Extension  of  the  jFormtila, 

74.  Let 

^  (.r)  dx  ~  (I),  {x), 


(f),  {x)  dx  =  (f)^^  (,r), 
etc.,     etc. ; 

then,  if  the  functions  (!)^  (x),  (f)^,  [x), ....  <f)„  [x),  which  may  be 
called  the  successive  integrals  of  (}>{x),  are  known,  and  also  the 
successive  derivatives  oif{x),  we  shall  have 

\f{x)  ^  (,r)  dx  =  f{x)  <!>,  (x)  -  |/'  (.r)  ^^v)  dx 

=  /(.r)  ^^  (.r)  -y  ■'(''^)  <f>^^  (x)  +  j/"  (.r)  ^,^  (x)  dx. 

Continuing  this  process,  and  writing  for  shortness  f,  <l>n  •  •  •  foi" 
f{x),  (f)^  [x)  .  .  .  we  have 


/{x)<f>{x)dx=/.<l>^-~/'.<^^^  + 


+   (-   \Y"f"~'(t)n 


+   {-lY      f"4n'dx. 

The  application  of  this  formula  is  equivalent  to  the  use  of  a 
formula  of  reduction.  Thus  the  value  of  x"^  £"^  given  in  Art.  65, 
may  be  derived  immediately  from  it. 


go  METHODS  OF  INTEGRATION.  [Art.  75. 

Taylor  s   Theorem. 

75.  If,  in  the  formula  of  the  preceding  article,  we  put 

f{x)  =F'  {Xo  +/i-  x),  and  <f>  (x)  =  i, 

Xo  and  /i  being  constants, 

/'  (x)  =  -  F"  {xo+  /i  -  x),        f"  {x)  =  F"'  {x,  ^  h-  x\  etc.  ; 

x^  x^ 

and        <f>,  (x)  =  X,       ^„  (.r)  =  — ,        <f>^,^  (x)  =  ^-^  ,  etc. 

Hence 

\F'{Xo+/i  —  x)dx=.F'  {xo  +  /i  —  x)'X-\-  F"  {xo  +  h  —  x)  -— 


+ 


F"^'  (Xo  +/i-  x)  — :^ dx. 


Now 


\F'  {xo  +  h  —  x)  dx  =  —  F  {Xo  +  h  —  x)  ; 


hence,  applying  the  limits  o  and  /i,  we  have 
Fix^  +  h)^  F{x,)  +  /' '  {Xo)  h  +  F"  {x:)  1-^  + 


Jo 


I-2-  •  ■  n 


This  formula  is  Taylor's  Theorem,  with  the  remainder  expressed 
in  the  form  of  a  definite  integral. 


VI.] 


EXAMPLES. 


91 


J  o 


Examples    VI. 


a:  sin"^  jc  +  v'(i  —  .v'^) 


7t 

2 


2.     sec~'jc</;c, 


tan~'  a;rfjvr. 


4- 


.V«  log  ,;c  ^jT, 


^sec-'^  —  log  [.V  +  -/(a"  —  i)], 

n      log  2 
4  2 

_;y«+i  r       _         I    " 

71  +   I  |_  °S  "^  —  ;;  +    ij  • 


5.        OsinQ^O, 


0  cos  iwO  I 


7.  Lv  tan-' a;</;c, 

8.  [x'a'^dx, 

J  o 

X  ^tc.-'^  X dx. 


It         I 

2?/2  /«^ 


I   +  X  X 

tan"'  X . 

2  2 


AT^f-*^  —  2^f-^  +   2  6-'^  —  2. 


I  [.r'  sec-'  X  —  -/(a*  —  i)]. 


IT 

o.    rosin    -  +  6   L/0,    — 0  cos(- +0)  +  sin( -+ e) 


TT  ^2 

4 


92 


METHODS  OF  IXTEGRATION.  [Ex.  VI. 


11.  \x  sec^  -v  dx^ 

12.  Atan^Ar</A-     =    Jc(sec^A-—  i)dx  |,xtan.x  +  log 

13.  \x^  sin  X  dxy 


X  tan  X  +  log  cos  x. 


cos  j: -v 


2A-  'jin  .V  4-  2  cos  a:  —  x  cos  a\ 


14.  AT  sin~ '  A- dTx,  -  A-  sin-'.v 

Jo  2  J^  2 

15.  v^  tan"  '  aVa, 


7T 

sin'  ^>  ^/O  =  -  . 

O 


■v'  tan  -  '  A-       .v'       log(i  +  .v^) 
3  6    "^  6 


16. 


■7T        2 


I  .  2    +   A 

-v'  sin-  '  A-  dx.    -  A-'  sin  - '  a-  +  ^  a/(i  —  a') 

3  9  J„        6       9 


17.        e-''  cosxdx. 


f  "-^  (sin  A-  —  cos.x)' 


18.     U  •" '^'"  ^  cos  a:  rt'AT, 


COS /3s^^^"P  sin  {ft  +  x). 


19.      f - ^  sin' .V ^/a-      =  -    f--^  (i  —  cos  2a:)^a:    , 

J  L      2  J  J 


(cos  2Ar  —  2  sin  2a:  —  5). 


I: 


20.        f*  sin  0  a'S, 


—  (sin  0  —  cos  0) 
2 


4  _  i_ 

2 


§VI.] 


EXAMPLES. 


93 


f^  sin  X  cos  X  dx, 
22.    I  sin^  ;«9  fl?i9, 


—  (sin  2x  —  2  cos  2X). 
lO  ^  ' 


sin^  m()  cos  ;;zQ        36        3  sin  we  cos  »«0 


4;// 


Zm 


23.  Derive  a  formula  of  reduction  for  Xi^ogxY  x"'  dx^  and  deduce 


from  it  the  value  of 


(log  x)"  X'"  dx  =  (log  x)"  ~ (log  x)"-^  X'"  dx. 


J(log.^•)^^-Va•  =  (logaf --  -  (logA)^--  + -^ 


24.      .V  cos^  A-  dx, 


27 


4 a"  sin x  cos .r  —  i  sin'  x  +  ^^. 


■'■I 


25.       jv-  sec"  '  a'^/jc, 


.r'  sec  -  ^  A-      .r  ^/(x''  —  i )       log  [x  +  V(.r'  —  i )] 


>6.  Derive  a  formula  of  reduction  for  Lv«  sin  {x  +  a)  dx,  and  de- 


duce from  it  the  value  of 


X  cos  X. 


X"  sin  {x  +  a)  dx  =  —  x'^sin  \  x  +  a  -] — 

+  n    a«"^  sir 
Lv'  cos  X  dx  =  {x''  —  20^'  +  120^)  sin  a:  +  {^x*  —  eox"^  +  120)  cos  x. 


7t 

X  -\-  a  ^r  - 
2 


94 


METHODS  OF  INTEGRATIO.W 


[Ex.  VI. 


f      -      .  ,      ,      sin' 0  COS  0      sin  OcosO   ,     i  r,        .    ^        ,-, 
27.     cos' 0  sin*  0  </i3,    7 T^ +  —  [0  — sinOcosOj. 


24 


16 


28.  *  cos*  0  sin*  0  ^0, 

•  o 

29.  COS 


i-Fsin*^V^'=^, 
32J0  S12 


Od^O, 


sine  cos' 0       isinOcosG      30' 
4  8  8. 


+  3^ 
32 


I 


30.         cos   0  ^G, 

sinO  cos  0  (8  cos*  0  +  10  cos' 0  +  15)  +  15G' 
48 


3  _  9'/3  +  i:>7r 
96 


J  sin'  e     ' 

J  cos  0      ' 

fsin'  G  ^ 
J  cos*  0 

flcos'o  ^ 
34-        -^ ^'^^ 


31 


32 


33 


cos'  ^J  _Z  cos  0      3  log  tan  4^ 
2  sin''  0  2  2 


sin  e 


4  cos  e       8  cos  0 


sin  0         I  , 

—  -log  tan 


L4       2  J 


sm  0         5  sm  0       t;  r  •  t 

i ^ —  +  ^  [G  —  sin  0  cos  0] 

3  cos   0        3  cos  0        2 


cos  0 

sin  0 


TT  IT 

2  f2 

-  5     cos 

IT  J  >r 


O^G 


48  —  IS^T 
32 


35 


36.  j 


^G 


+  cos  0)' 

^0 


sin  G  cos  G 


4     d^)'     _  2 
cos*  G'       3  * 


5-  H +  log  tan  -  . 

3  cos  0       cos  G  2 


§VI.] 


EXAMPLES. 


95 


37 


J  sii 


M 


sm  6  sin   20  4  sm  Q  cos 

38.  Prove  that  when  n  is  odd 


3  cos  63,  0 

^    ■  o      +  5  logtan- 
8  sin  e       8      ^        2 


J  six 


^9         _sec«-^0       sec«-5G 
I  sin  6  cos"  0        n  —  X  //  —  3 

and  when  ;2  is  even 


da 


sec"-^0       sec"-^© 

-I 1_ 


I  sin  0  cos''  0        n  —  1  n  —  2, 


+  log  tan  S  ; 


+  log  tan  ■ 


Z9-    \- 

■         J  si 


do 


40. 


41- 


^ -+5 

sin"'&cos*0'        2  sin"  6  cos'^O       2 


'sec  0 
h  sec  0  +  log  tan 

L     3  2 


] 


V'l^v^-  i) 


,     Put  X  =  sec 


U-y(a-^  -  i)  +  ilog  [.r  +  Vix'  -  i)]. 


{a'  —  x')'  dx,  ^ '—-^ ^  +  iL_  sm  - 1  - 


cos*  0  ^0  =  ^ r-  . 


44- 


45- 


J(-^-^  +  i)" 


V{x'  -  a') 


+  log  [x  +  ^{x'  -  a')]. 


x{x^  —  1)    ,  tan"'ar 

8(x'  +  iy^~8    ■ 


96 


METHODS  OF  INTEGRATION.  [Ex.  VI. 


,     fcos'G  — sin'O    j,V[i,-,         ,\    cosO  — sinO 

46.    L  . vJ '^'^      =     (i  +  sin  0  cos  J/);^ 

^      J(sinG  +  cos  6)'        L      J 


,^0], 


(sinO  +  cosO) 

sin  0  cos  5 
sine  +  cosO 


47.  Derive  a  formula  for  the  reduction  of  \x%Q.z^xdx\  and  refer- 
ring to  Ex.  II,  thence  show  that  this  is  an  integrable  form  when  n  is 
an  even  integer.     Give  the  result  when  //  =  4. 


-v  sec"  -v  dx  = 


a-  sec"  -  ^  -v  tan  x  sec"  -  ^  :>; 


«  —  I 


X  sec*  X  dx  = 


{n-i){n-  2) 

n  —  I  J 
X  sec*  X  tan  x      sec'  x      2 


V  sec"-''x  dx. 


^-. — h  -  [-v  tan  .V  +  log  cos  .rl. 
6  3 


48.  Derive   a  formula  of   reduction  for 
from  it  the  value  of    v  cos^  x  dx. 


X  cos"  X  dx,  and  deduce 


,         .vcos""' .V  sm  .V       cos"-v      «—  i  f 

.V  cos"  -v  dx  =  H r. 1 .r  cos"  ~  ^  .V  dx. 

n  n  n      ) 


3       ,         .vsin.r.       ,  .        cos.r. 

-v  cos  -v  dx  = (cos  X  +  2)  -\ (cos-  X  +  6). 

3  9 


49.   Find  the  area  between  the  curve 
_y  =  sec  " '  .r, 


§  VI.]  EXAMPLES.  97 

the  axis  of  .v,  and  the  ordinate  corresponding  to  .v  =  2. 

27C 

—  -  log  [2  +  rj]  =  0.77744. 

50.  Find  the  area  between  the  axis  of  .v,  the  curve 
y  =  tan" '  -v, 

77"         l0£  2 

and  the  ordinate  corresponding  to  x  —  \.  e—  =  0.43882. 


VII. 

Definite  Integi'als. 

76.  Before  proceeding  to  transformations  of  definite  inte- 
grals  involving  the  values  of  the  limits,  it  is  necessary  to 
resume  the  consideration  of  the  relations  between  a  definite 
integral  and  its  limits,  as  defined  in  the  first  section. 

By  definition,  the  symbol 

•  a 

denotes  the  quantity  generated  at  the  rate 

while  X  passes  from  the  initial  value  a  to  the  final  value  X. 
The  rate  of  x  is  arbitrary,  and  may  be  assumed  constant  ;  but 
in  that  case  its  sign  must  be  the  same  as  that  of  the  mcrement 


98  METHODS  OF  INTEGRATIOX.  [Art.  76. 

received  by  .r ;  that  is,  the  sign  of  dx  is  the  same  as  that  of 
X-  a. 

These  considerations  often  serve  to  determine  the  sign  of 
an  integral.     Thus 

f"  sin  xdx 


.       ,                 ,    .           .  .             ,  sm  X 
denotes  a  positive  quantity,  because  dx  is  positive,  and   

is  positive  for  all  value;;  of  x  between  o  and  n. 

77.  Now  let  F  {x)  denote  a  value  of  the  indefinite  integral, 
so  that 

d{F{x)\=fKx)dx', 

thus/(,r)  is  the  derivative  of  F{x).  Then,  supposing  F  (x)  to 
vary  continuously  as  x  passes  from  a  /^  X ;  that  is,  to  have  no 
infinite  or  imaginary  values  for  values  of  x  between  a  and  X^ 
the  integral  is  the  actual  increment  received  by  F  {x),  while  x 
passes  from  a  to  X.     In  this  case,  therefore 

\^'f{x)dx  =  F{X)-F{a) (I). 

If,  on  the  other  hand,  there  is  any  value,  n-,  between  a  and  X, 
such  that 

F{a)=.^, 

equation  (i)  does  not  hold  true.     For  example, 

[dx  _ 


[dx  I 

X 


and  in  the  case  of  the  definite  integral 


dx 
1? 


§  VII.]  DEFINITE  INTEGRALS.  99 

X  passes  through  the  value  zero,  for  which  F  (x)  is  infinite ;  %ve 
c CHI  not  therefore  tvrite 


dx  __       I 

Z2    ~ 


X 


~  —  2. 


This  result  indeed  is  obviously  false,  since  dx  is  here  positive, 
and  x^  is  never  negative  for  real  values  of  x.  The  value  of  the 
integral  is  in  fact   infinite,  since  the    increments  received    by 

,  while  x  passes  from  —  i  to  o,  and  while  x  passes  from  o 

to  I,  are  both  infinite  and  positive. 

78.  Since  the  derivative  of  a  function  becomes  infinite  when 
the  function  becomes  infinite  [Diff.  Calc,  Art.  104;  Abridged 
Ed.,  Art.  89],  we  can  have  F  {a)  =  00  only  when  /{a)  =  00; 
but  it  is  to  be  noticed  that  Fix)  does  not  necessarily  become 
infinite  wh.en/{x)  becomes  infinite.     Thus,  in 

f'  dx 

/{x)  —x-^i,  which  becomes  infinite  for  ;r  =  O,  a  value  of  x 
between  the  limits  ;  but  since 

Lr-7  dx  =  |;r^ 

the  indefinite  integral  F{x)  does  not  become  infinite.  There- 
fore equation  (i)  holds  true,  and 

dx       3    „-l3         9 

.      ,a4  2        J_,         2 

79.  We  have,  in  the  preceding  articles,  assumed  that  the 
independent  variable  varies  uniformly  in  passing  from  the 
lower  to  the  upper  limit ;  but  when  a  change  of  independent 
variable  is  made,  the  new  variable  does   not   generally   vary 


100  METHODS  OF  IXTEGRATIOX.  [Art.  79. 

uniformly  between  its  limits.  It  is,  however,  obvious,  that,  in 
equation  (i),  Art.  ']'J^  x  may  vary  in  any  manner  whatever  in 
passing  from  a  to  A',  provided  tliat  F{.r)  remains  tliroughotit 
a  continuous  onc-valucd  function ;  x  may  even  pass  through 
infinity,  provided  F {x)  is  finite  and  one-valued  when  .«■  =  00  . 


Mtdtiplc-  Valued  Integrals. 

80.  When  the  indefinite  integral  is  a  multiple-valued  func- 
tion, a  particular  value  of  this  function  must  of  course  be 
employed,  and  it  is  necessary  to  take  care  that  this  value  varies 
continuously  while  x  passes  from  the  lower  to  the  upper  limit. 
In  the  fundamental  formula  (_;')  it  is  sufficient  (provided  the 
radical  t(i  —  x^)  does  not  change  sign),  to  limit  the  meaning  of 
the  symbols  sin~'-r  and  cos" 'a-  to  the  primary  values  of  these 
symbols  (see  Diff.  Calc,  Arts.  54  and  55),  since  these  values 
are  so  taken  as  to  vary  continuously  while  x  passes  through 
all  its  possible  values  from  —  i  to   f  i. 

81.  In  the  case  of  formula  {k)  the  primary  value  of  tan"'  x 
is  so  defined  that,  as  x  passes  from  -co  to  +  00  ,  the  primary 
value  varies  continuously  from  —  |^rr  to  +  l^r.  We  may  thcre- 
fo're  employ  the  primary  value  at  both  limits,  unless  x  passes 
through  infinity^  as  in  the  following  example.  Given  the  inte- 
gral 

f?  de  [i  ^cc^Odfi 


< 


]^cos^d  -\-gs\n^d      J^i  +  Qtan^d*' 
if  we  put  tan  6  =  .r,  this  becomes 


"  3       dx  I  ^ 

5-  =  -  tan -'3.1- 

i  +  <^       3 


=  -  [tan-'(-  «  3)  —tan-'  o]. 


J 


But  here  it  is  to  be  noticed,  that,  as  6  passes  from  o  to  ^rt,  x 


§  VII.]  MULTIPLE-  VALUED  JJ^^ECMLt^  ]  ]  -/  ] \  >'  \  /\'^^^ 

passes  through  infinity  when  6  =  |7r.  Hence,  if  the  value  of 
tan~'3,i'  is  taken  as  O  at  the  lower  limit,  it  is  to  be  regarded  as 
increasing  and  passing  through  Itt,  when  ^'=00,  so  that  its 
value  at  the  upper  limit  is  f  tt,  and  not  —  ^tt.     Hence 

f6  d6  2n 


Jo  cos^  6^  +  9  sin^  6       g  ' 

82.  When  the  symbol  cot-^  x  is  employed,  the  primary 
value,  defined  in  the  same  manner  as  in  the  case  of  tan~'x, 
cannot  be  taken  at  both  limits  when  x  passes  through  zero. 
Thus,  using  the  second  form  of  (-^),  Art.  10,  we  have 


!.: 


=  cot-'  I  —  cot-'(—  l), 


+  ^ 


in  which,  if  cot"'  i  is  taken  as  -]  ;r,  cot-'(—  i)  must  be  taken 
as  I  7t.     Thus 


dx  I 
-„  =  —  -  ;r. 


I    I  ^  x" 


Formulas  of  Reduction  for  Definite  Integrals. 

83.  The  limits  of  a  definite  integral  are  very  often  such  as 
to  simplify  materially  the  formula  of  reduction  appropriate  to 
it.     For  example,  to  reduce 

x^  e  -"^dx. 


we  have  by  the  method  of  parts 

{x"s-^dx  =  —  x"*  £-^  +  n  U--'' x"-'dx. 


102    ',  I  i/f  ',',:  '■'  ''yMETHOJlS.  QF  INTEGRATIOK.  [Art.  83. 

Now,  supposing  n  positive,  the  quantity  ;«•'' f""^  vanishes  when 
;jr  =  o,  and  also  when  a'  =  00  [See  Diff.  Calc,  Art.  107 ;  Abridged 
Ed.,  Art.  91].     Hence,  applying  the  limits  o  and  00, 


Jo  Jo 

By  successive  application  of  this  formula  we  have,  when  n  is 
an  integer, 

x"  £-■'' dx  =  n  {h  —  i) 2  •  I . 

J  o 

84.  From  equation  (i)  Art.  66,  supposing  ;n  >  i,  we  have 
f'  sin'«  edd  =  ^.^^-^^  \"  sin"'--  OdS. 

Jo  ^«        Jo 

If  m  is  an  integer,  we  shall,  by  successive  application  of  this 


fl  7t 

formula,  finally  arrive  at       dO  =  -  or 

Jo  2 

as  7n  is  even  or  odd.     Hence 


sin  6 dd  =  I,  according 


if  7n  IS  even,  sm'"  0  dd  =  ^ ^-^ ^-^ ,  .  .  (P) 

J^  m{m  ~  2) 22'         ^    ' 

and  if  m  is  odd,       p  sin'"  0  dd  =  ^''' ~ /X'l^Il^hlUL^ ,  •  -  {P') 
Jo  m{m  —  2) I 


§  VII.]  FORMULAS  OF  REDUCTION.  IO3 

•85.  From  equations  (3)  and  (4)  Art.  69,  we  derive 

sin"'(9cos"-^/9^«9, 


sin'«  e  cos"  BdB= ^ 

m  +  ;/  . 


and         I '  sin'«  B  cos"  B  dB  =  ^^  ~  ^    V  sin'«-^  B  cos"  B  dB. 

Jo  ^^l-    +    ^^    Jo 


By  successive  application  of  these  formulas,  we  shall  have  for 
the  final  integral  one  of  the  four  forms 


dB, 


s'm  B dB,        rcos^<^6',  or  |    sin^cos^^^. 

Jo  Jo 


The  numerator  of  the  final  fraction  ( or )  is  in  each 

\m  +  n       ni  +  71 J 

case  either  2  or  i.  In  the  first  case,  the  value  of  the  final  inte- 
gral is  I  TT,  and  the  final  denominator  is  2 :  in  the  second  and 
third  cases,  the  value  of  the  final  integral  is  i,  and  the  final 
denominator  is  3  :  in  the  fourth  case,  the  value  of  the  final 
integral  is  \,  and  the  final  denominator  is  4.  Therefore  (since 
the  factors  in  the  denominator  proceed  by  intervals  of  2),  it  is 
readily  seen  that  we  may  write 


Fsin"  e  COS'.  Ode  =  («-)('«- 3)- ■•■('' -0(>'- 3)  ■■•,,  .  (g) 
j^  {m  +  n){m  +  n  —  2) 

provided  that  each  series  of  factors  is  carried  to  2  or  i,  and  a  is 
taken  equal  to  tmity,  except  when  m  and  n  are  both  even,  in  which 
case  a  =  ^TT. 


104  METHODS  OF  INTEGRATION.  [Art.  86. 


Elementary  Theorems  Relating  to  Definite  Integrals. 

86.  The  following  propositions  are  obvious  consequences  of 
equation  (ij,  Art.  77. 


^f{x)dx^-^fi,x)dx (I) 


\  f{x)dx=\  f{x)dx  +\  f{x)dx 

Ja  *  a  i  c 

Again,  if  we  put  x  —  a  +  b  —  s,\\q  have 

fV W dx=-  \^f{a  +  b-z)dz=( f{a  +  b- z) 

ia  Jb  ia 


.      .      (2) 


dz 


by  (i),  or  since  it  is  indifferent  whether  we  write  5-  or  ;r  for  the 
variable  in  a  definite  integral, 

^f{x)dx=    (f{a-Vb-x)dx     ....     (3) 

If  «  =  c,  we  have  the  particular  case 

^f{x)dx=tf{b-x')dx     ....     (4) 


§  VII.]  DEFINITE  INTEGRALS.  IO5 

87.  As  an  application  of  formula  (4),  we  have 

JT  77  IT 

J'  cos"'  Bde^^  cos"Y|  -  B\  dO  =  y  sin"'  Odd     .     .     .     .     (i) 


value  of 


Hence  the  value  of      cos'"  0  dd  as  well  as  that  of 


sin'«6^^/^ 


is  given  by  formulas  (P)  and  (P).  The  values  of  these  integrals 
are  readily  found  when  the  limits  are  any  multiples  of  A  rr. 
For,  by  equation  (2)  of  the  preceding  article,  we  may  sum  the 

7t 

values  in  the  several  quadrants.  But,  putting  6  —  k — h  6',  and 
employing  equation  (i),  we  have 

1^    'sin«^^^=±|        '  cos-^^^=  ±]%in'«6'^6',    ...    (2) 


/.— 


in  which  the  sign  to  be  used  is  determined  by  that  of  sin"  0 
or  cos"'  6  in  the  given  quadrant. 

In  like  manner  the  value  of  the  integral  in  formula  (Q)  is 
numerically  the  same  in  every  quadrant,  and  its  sign  is  the 
same  as  that  of  sin"'  6  cos"  6  in  the  given  quadrant. 


Change  of  hidependent  Variable  in  a  Definite  hitegraL 

88.  It  is  often  useful  to  make  such  a  change  of  independ- 
ent variable  as  will  leave  unchanged,  or  simply  interchange, 
the  values  of  the  limits.  As  an  illustration,  let  us  take  the 
definite  integral 

■"       log.r        , 
IL  =       2 dx. 

)ol   +  X  -^  X^ 


io6 


METHODS  OF  INTEGRATIOiW 


[Art.  88 


If  we  put  X  —  - ,  whence  log  x  =  —  log  j,  and  cix  =  — 


7 


r       logy 
u  =       -,         — 


<yi'  —  —  2i\ 


whence  we  infer  that 


]o\    ^  X 


\ogx 


+  ;i-^ 


-xuix  —  o. 


89.  Again,  let 


"  = '  <?T?''-"- 


/ 


Putting  A-  =  —  ,  we  have 


hence 


dv 


d^  +  f 


2  log^  —  log  1/     ,  , 
2_ ^.rl.  dy  =  2   log  <7 


r]og^^_^^7t\oga 


Differentiation  of  an  Integral. 


90.  The  integral 


f  {x)  dx  is  by  definition  a  function  of  .r, 


whose  derivative,  with  reference  to  x,  is  f{x).     Thus,  putting 


dU 
dx 


f{x)dx, 
=  f{x). 


§  VIL]  DIFFERENTIATION  OF  AN  INTEGRAL.  10/ 

This  gives  the  derivative  of  an  integral  with  reference  to  its 
upper  limit.     By  reversing  the  limits  we  have,  in  like  manner, 

dV  ,,  , 

v;hen  the  lower  limit  is  regarded  as  variable. 
9L  Now  writins:  the  integral  in  the  form 


U^ 


II  dx , (i) 


if  71  is  a  function  of  some  other  quantity,  n-,  independent  of  x 
and  a,  U  is  also  a  function  of  <t,  and  therefore  admits  of  a  de- 
rivative with  reference  to  a.     From  (i)  we  have 

dU 

whence 

d  dU  __  dn 
da  dx        doc 

By  the  principle  of  differentiation  with  respect  to  independent 
variables  [See  Diff,  Calc,  Art.  401 ;  Abridged  Ed.,  Art.  200]. 


Therefore 


and  by  integration 


d^  dU _   d^  (W 
dx  da       da  dx 

d  dU  _  dzi  ^ 
dx  da       da ' 


dU      [du 
da 


\fj..C    ......(.) 


I  OS  METHODS  OF  INTEGRATION.  [Art.  9 1. 

Now,  in  equation  (1),  ^/ is  a  function  of  x  and  ex  which,  when 
X  =  «■,  is  equal  to  zero,  independently  of  the  value  of  a.  In 
other  words,  it  is  a  constant  with  reference  to  a,  when  x  —  a\ 

therefore    -- —  o  when  x  =  ^.     If,  then,  we  use  rt;  as  a  lower 
dix 

limit  in  equation  (2),  we  shall  have  ^7=0.     Therefore 
dU      [   du 


da       ).dJ-' (3) 

Substituting  for  x  any  value  b  independent  of  «-,  we  have 

\    udx^\    ~~udx, (4) 

a  a  it  J  a  da 

which  expresses  that  a7i  integral  may  be  differentiated  luith 
reference  to  a  quantity  of  ivJiicJi  the  limits  are  independent,  by 
differentiating  the  expression  Jinder  the  iiitegral  sign. 

92.  By  means  of  this  theorem,  we  may  derive  from  an  inte- 
gral whose  value  is  known,  the  values  of  certain  other  inte- 
grals.    Thus,  from  the  first  fundamental  integral, 

\x"dx  =  - , (i) 

we  derive,  by  differentiating  with  reference  to  n, 

Lr"  log  X  dx  =  ^ '- \ , 

the  result  being  the  same  as  that  which  is  obtained  by  the 
method  of  parts. 

93.  The  principal  application  of  this  method,  however,  is 
to  definite  integrals,  when  the  limits  are  such  as  materially  to 


VIL] 


DIFFERENTIATION  OF  AN  INTEGRAL. 


109 


simplify  the  value  of  the  original  integral.     Thus,  equation  (i) 
of  the  preceding  article  gives 


(•1  T 

X"  dx  — 

Jo  '^+1 


whence,  by  successive  differentiation, 

I 


;f ''  loSf  X  dx  =^ 


{n  +  if 


x""  (log  xf  dx  = 


1-2 


{n  +  if 


x"  (log  xf  dx  =  {— if 

J  o 


1-2 


(n  +  1)''  +  ^ 


Integration  under  the  Integral  Sign. 

94.  Let  u  be  a  function  of  x  and  «-,  and  \e±a  and  a^  be  con- 
stants ;  then  the  integral 


U. 


u  dx  \day (l) 


is  a  function  of  x  and  a,  which  vanishes  when  a  =  a^,  inde- 
pendently of  the  value  of  x,  and  when  x  =  a,  independently  of 
the  value  of  a.     From  (i) 


dU 


dU      f        , 
— -=       udx, 
da       J  a 


therefore      -—-—=:?/, 
dadx 


whence 

/ 

whence 


d_dU_ 
dx  da 


-z-  =  \ii  da  +  6. 


HO  METHODS  OF  INTEGRATION.  [Art.  94. 

Now-i—  must  vanish  when  a  =  a-^,  since  this  supposition  makes 

t/ independent  of  x\  therefore,  if  we  use  «"„  for  a  lower  limit 
in  the  last  equation,  we  must  have  C  =  0\  therefore 

-J-  =       21  da, 
ax        J  a^ 

and  since  u  vanishes  when  x  =  a, 

U^lllud.'ld.. (3) 

Comparing  the  values  of  6^  in  equations  (i)  and  (2), 'we  have 


It  dx  doc  =  \     \    u  da  dx. 


It  is  evident  that  we  may  also  write 

r«i  [b  tb  rcxi 

u  dx  da  =  71  da  dx,     .     .      .     .     (i) 


provided  that  the  limits  of  each  integration  are  independent 
of  the  other  variable. 

95.  By  means  of  this  formula,  a  new  integral  may  be  de- 
rived from  the  value  of  a  given  integral,  provided  we  can  inte- 
grate, with  reference  to  the  other  variable,  both  the  expres- 
sions under  the  integral  sign  and  also  the  value  of  the  inte- 
gral.    Thus,  from 

I    x"  dx  = 


n  +  I 


VII.]      INTEGRATION   UNDER    THE  INTEGRAL   SIGN.         Ill 


by  multiplying  by  dn,   and  integrating  between  the  limits   7 
and  s,  we  derive 

X"  dn  dx  =  I    , 

Jo   J.  J./^+     I 


whence 


'  X'  —  x""     ,         .       s  +  I 
dx  =  lop- 


loo-  X 


r  +  I 


96.  When  the  derivative  of  a  proposed  integral  with  refer- 
ence to  «  is  a  known  integral,  we  can  sometimes  derive  its 
value  by  integrating  the  latter  with  reference  to  a.     Thus,  let 


u  = 


">c-o.r 


—  s- 


dx. 


In  this  case 


du       r  ^        s-^ 


hence,  integrating,        u=—  log  a  +  C  =  log  — 
since  in  (i)  u  vanishes  when  a  =  0. 


(I) 


.    (2) 


The  Definite  Integral  Regarded  as  the  Limitmg  Value 

of  a  Sum. 


97.  Let  A  denote  the  greatest,  and  B  the  least  value  as- 
sumed by/"(a-),  while  x  varies  from  a  to  b.  Then  it  is  evident 
that 


\f{x)dx< 

J  a 


A  dx\ 


(0 


for,  while  x  passes  from  a  to  b,  the  rate  of  the  former  integral 


112  METHODS  OF  INTEGRATION.  [Art.  97. 

is  generally  less,  and  never  greater  than  the  rate  of  the  latter. 
In  like  manner 

(  f{x)dx>  \!'  Bdx (2) 


The  values  of  the  integrals  in  the  second  members  of  equations 
(i)  and  (2)  are  A  {b  —  a)  and  B  {b  —  a)  respectively.  There- 
fore, if  we  assume 

^f{x)dx  =  M{b-a), (3) 

we  shall  have  A  >  M>  B. 

The  quantity  M  in  equation  (3)  is  called  the  mean  value  of  the 
function /"(-t-)  for  the  interval  between  a  and  b. 
98.  Let 

b  —  a  =1  n  rix\ (4) 

then  the  w  +  i  values  6f  x, 

a,  rt  +  A  ;r,  rt:  +  2  A  ;f ,  •  •  •  •         b, 

define  n  equal  intervals  into  which  the  whole  interval  b  —  a  \s 
separated.  Let  Xi,  Xn, x„he  n  values  of  x,  one  com- 
prised in  each  of  these  intervals;  also  let  2^/{Xr)  Ax  denote 
the  sum  of  the  n  terms  formed  by  giving  to  r  the  n  values 
1  •  2  •  •  •  •         7/  in  the  typical  term/(-tv)  Ax;  that  is,  let 

2i  f{Xr)  AX  =  f{x,)  A 4.-  +  /(xg)    AX"'-  +  f{x„)  A X.    .     .    (5) 


§  VII.]  AN  INTEGRAL    THE  LIMIT  OF  A    SUM.  1 13 

We  shall  now  show  that  when  n  is  indefinitely  increased  the 
limiting  value  of  2if{xr)  Aa'  is     f{x)  dx. 

99.  If  we  separate  the  integral  into  parts  corresponding  to 
the  terms  above  mentioned  ;  thus, 

eh  C'i  +  A-r  fa  +  2i^x 

f{x)dx^  f{x)dx+  f{x)dX'-.' 

J  a  ■^  ''  Ja  -i-  L  .1- 


+ 


f{x)  dx, 


b  -    A. 


and  let  l/j,  M^,  •  •  •  •  Mn  denote  the  mean  values  of  /(,r) 

in  the  several  intervals,  we  have,  in  accordance  with  equation 
(3),  Art.  97, 

[  fix)  dx  ~  Mi^x  ■^M^t.x +  Mn  i\x (6) 


Now,  since  f{x,)  and  M,.  are  both  intermediate  in  value 
between  the  greatest  and  the  least  values  oi  f  {x)  in  the  inter- 
val to  which  they  belong,  their  difference  is  less  than  the  dif- 
ference between  these  values  oi  f{x).     Therefore,  if  we  put 

f{Xr)^AIr   +   er, (/) 

er  is  a  quantity  whose  limit  is  zero  when  n,  the  number  of 
intervals,  is  indefinitely  increased,  and  i\x  in  consequence 
diminished  indefinitely. 

Comparing  the  terms  in  equations  (5)  and  (6)  we  have,  by 
means  of  equation  (7), 

:S*/(a')  ax  =  [  /(-^')  dx  +  {e^-\-  e^ +  e„)  Ax.      ...      (8) 


114  METHODS  OF  INTEGRATION.  [Art.  99. 

Denote  by  e  the  arithmetical  mean  of   the  «  quantities  rj, 
^2.  •  •  •  •  ^n ;  that  is,  let 


«r  =  ^1  +  rj  +  rg r„  ; (9) 


then,  ^ince  e  is  an  intermediate  value  between  the  greatest  and 
the  least  value  of  Cr,  it  is  also  a  quantity  whose  limit  is  zero 
when  11  is  indefinitely  increased.  By  equations  (9)  and  (4), 
equation  (8)  becomes 


^\f{x,)  A.t-  =  f  f{x)  dx  +  e{b-  a), 


whence  it   follows  that 


/(.r)  dx  is  the  limit  of  -^„/  (x,)  dx 


when  n  is  indefinitely  increased,  since  the  limit  of  c  is  zero. 

100.  It  was  shown  in  the  Differential  Calculus,  Art.  390 
[Abridged  Ed.,  Art.  193],  that,  in  an  expression  for  the  ratio 
of  finite  differences,  we  may  pass  to  the  limit  which  the  ex- 
pression approaches,  when  the  differences  are  diminished  with- 
out limit,  by  substituting  the  symbol  d  for  the  symbol  a. 
The  theorem  proved  in  the  preceding  articles  shows  that,  in 
like  manner,  in  the  summation  of  an  expression  involving 
finite  differences,  we  may  pass  to  the  limit  approached  when 
the  differences  are    indefinitely   diminished,  by  changing    the 

symbols  -2"  and  A  into     and  d. 

The  term  integral,  and  the  use  of  the  long  s,  the  initial  of 
the  word  sum,  as  the  sign  of  integration,  have  their  origin  in 
this  connection  between  the  processes  of  integration  and  sum- 
mation. 


VII.]      ADDITIONAL  FORMULAS  OF  INTEGRATION.  II5 


Additional  FoJ'imdas  of  Integration. 

10!.  The  formulas  recapitulated  below  are  useful  in  evalu- 
ating other  integrals.  {A)  and  {A')  are  demonstrated  in 
Art.  17;  {B)  and  {C)  in  Art.  29;  {D)  and  {E)  in  Art.  30; 
{F)  in  Art.  31  ;  (Q  and  {G')  in  Art.  35  ;  {H)  and  (/)  in  Art.  50 ; 
{J)  in  Art.  51  ;  (A')  in  Art.  52  ;  (Z)  in  Art.  53  ;  (J/)  in  Art.  55  ; 
(A")  and  {O)  in  Art.  58;  {P)  and  {P')  in  Art.  84;  and  (0  in 
Art.  85. 


dx 


loc 


(.1-  -  a)  {x  -  b)       a  -b     "=  X  ~b 

dx  I  ,      X  —  a 

lot 


2a 


'X  -^  a 


s\v?  6  do  -  l{8  -  sin  6  cos  0) .     .     .     .     . 
9  =  |(^  +  sin  ^ cos  ^) .     .     o     o     . 


COS' 


o'^ 


sin  6  cos  ^ 


losf  tan  0 


-,  n       1       I  ~  cos 

.     ^  =  log  tan  4c/ =  log r — -— 

sm  ^  ^  ^  ^      sm  ^ 


^6* 
cos  0 

dd 


=  log  tan 


TT 

^~ 

+ 

— 

U 

2  _| 

=  losf 


I  +  sin  ^ 
cos  6 


+  6  cos  6       V{a^  -  ^) 


tan" 


/  7  tan  4i 

a  +  b 


II J 


METHODS  OF  INTEGRATIOX.  L^l't.   lOI. 


dd 


,       Vi^  +  a)  4-  V{d  —  a)  tan  J  ^  ,^,. 


\a+dcose      Vip'-a^)     ^  ^/[b^a)  -  V{d  -  a)  tan  ^  6 


dx 


1 ,1-  v^  {x^  +  rt'-^)       rt: 


—  -  \oz  -^ 


(^j 


dx 


1   ,       a  —  \'(c^  —  x^) 

-  lor 


X  \/{c^  -^)       a      "^  X 


(/) 


dx 


^{ax'-^bf-      bV{a^^+  b) 


{y) 


1 


dx 


Vi-r"  ±  a") 


log  [_x  +  i/(.t-2  ±  ^)] 


(>^) 


1 1/(;.^  ±  a')  dx  =  ^^(-^''^^^')   ±  l'  log  [x  +  VU""  ±  ^=)]  •    •    (L) 


I 


.r      ;f  Vf^z^  —  .v^ 


s/  id}  —  x^)  dx  —  —  sin  - '  -  + 
^  '  2  a  2 


(J/) 


dx 


^\{x-a){x-p)\ 


=  2  log  [  V(-r  -  ^0  +  i/f.r  -  ^^]  .     .     .    {N) 


dx 


^l{x- a)^li  -  x)\       ^''^"^    li 


(0 


sin'"  e  do  = 


m{m  —  2) 2    2 


§  VII.]      ADDITIONAL  FORMULAS  OF  INTEGRATION.  WJ 


sin"'  6  dB  = 


cos^'Odd^"^ ^^ ^^' .     .     .    {p'\ 

in  {in  —  2) I  ^     ' 


sin' 


6  cos"  Odd  —  ^ ^^  ,,   -^^ ^^ — r — ^-^ ^ a,     .  (0) 

{in  +  n){in  +  n  —  2) ^""^ 


in  which  a  =  i,  unless  m  and  n  are  both  even,  when  a  = 


Examples  VII. 


a  +  d  cosQ' 


[a  >  d,  and  n  an  integer] 


nir 


V{d'  -b') 


2. 


•2«7r±-  ^g 


2    +  COSS' 


2n7C±\7t 


'■1 


3.        sm  Ode, 


55 
32" 


4- 


sin^  0  do, 


16 
15 


„  cos"  QdQ, 


16 

35' 


!>' 


6.      sin^  0  cos^  Q  (/9, 


37r 


ii8 


METHODS  OF  INTEGRATION.  [Ex.  VII. 


7.  sin'  0  cos^  6  ih. 

It 

8.  I   sin"'  0  cos'"  0  ^/G, 

f      X-"  dx 

x^i^a'  —  x^y  dx, 


1 

-P 

2'"Jc 


3_ 
35 


sin"'  0  r/O , 


1-3-5  •  • 

(2«  — l)    TT 

2-4-6  •   • 

■    2n        2 

2-46 

•    •  2/1 

3-5-7-  • 

•    (2;/  +1)* 

2a' 
^3 

i6a ' 

13- 


14 


.r^  dx 


.v'  dx 


15.  Prove  that 


TT 

12a 


.v''-'(a  — -r)"'-'  dx  —  I    .v'«-'  {a  —  x)"--"  dx , 

Ju  o 

and  derive  a  formula  of  reduction  for  this  integral,  supposing  «  >  0 
and  m  >  i. 

It                                        ffi —  if'* 
X" -^  {a  —  x) '" - '  dx  =  A"  {a — x) '" -^  dx. 


§  VII.l 


EXAMPLES. 


119 


16.  Deduce  from  the  result  of  Ex.   15   the  value  of  the  integral 
when  7n  is  an  integer. 


X"  - 1  {a  —  x) '«  -■"  dx  = 


•J  \  '  ^m  +  n-i 


17- 


n{n  +  i)   •    •    •{n  +  m  —  i) 

a 
(a  +  xy  (a  —  x-y  dx.     See  Ex.  16. 

a 

8.       sin'  0  (cos  ^y  do.     Put  sin^  Q  =  x,  a?td  see  Ex.  16. 


a  *  , 

45045 


5-7-II-I9 


19.  Show  by  a  change  of  independent  variable  that 


x'^  dx 


dx 


and  therefore 

\x  log  A-  dx 


o  {a'  +  xy  -Jo  {a'  +  xy  ' 

f"'       X'  dx        _  £  r       dx        __   7t_ 
Jo    (a"  +  X'y  ~  2]od'   +  .V'~  4a' 


■  Jo    {x'+ay  ' 

rtan-^v.  dTjc 

■  Jo^*  +  ^  +  i' 


'•i; 


tan     — 


x  dx 


4       I        _4  > 


4        I  • 

a   X  +  a 


logo. 
2a" 


6V3- 
i6d'' 


23.  Derive  a  series  of  integrals  by  successive  differentiation  of  the 
definite  integral  |    ?-"•''  dx. 


X"  €-'^  dx  = 


I-2-  •  •  n 


120  METHODS  OF  INTEGRATION.  [Ex.  VII. 

24.  Derive  from  the  result  of  Art.  63  the  definite  integrals 

£- ""^ %\Xi  nx  dx  = —f-^ — »,     and  i- "'"^ 0.0%  nx  dx  = —^ — ^\ 

Jo  m   +  n  '  Jo  ///    +  n 

and  thence  deri  e  by  differentiation  the  integrals     ' 

|"00  -3i  3  3 

f     „    ,„,  .  ,  2w,v  ,1  ,  m  —  n 

xe-  "'' sin  fix  dx  —  -~r. ij-. ,  and      xe -  '«■*  cos  nx  dx  =  t^, rr» ' 

Jo  (/«■  -;-  11)  Jo  (;«'  +  r  ) 

25.  From  the  results  of  Ex.  24  derive 

X'  €-  """  sin  ?2X  dx  =  -~^. J—  : 


|"W  / 3  _  ..3 

x''  e-  "'-^  cos  nx  dx 


26.  From  the  fundamental  formula  {H^)  derive 


and  thence  derive  a  series  of  formulas  by  differentiation  with  refer- 
ence to  oc. 


r'        dx  _     ^      i-3'"*(2«  — 3)       I 

Jo(«  +y».vy  ~  ^ri-2  •••(«-  !)■««-* 


27.  Derive  a  series  of  integrals  by  differentiating  with  reference  to 
yS,  the  integral  used  in  Ex.  26. 


f"    X^*^-^  dv     _      7T     I-3-5  •  •  (2«  —  3)         I 
Jo  (a  +  fix^Y  ~  ^^  1.2-3  •  •  •  («-  i)  *  /:;''"*  ' 


§  VII.] 


EXAMPLES. 


121 


28.  From  the  integral    employed   in  examples  26    and  27,   derive 

x"  dx 


the  value  of  ,     ,  ,^  ,,, 

Differentiate  tiaice  with  reference  to  /3,  and  once  ivith  reference  to  a. 

p       X*  dx         _  i'3-i  7t 

Jo  {a  +  l±xy  ~  TTa^  ■  ^eaifi^  ' 

29.  Derive  an  integral  by  differentiation,  from  the  result  of  Ex.  II.,  67. 

dx  _     7T  {2a  +  b) 

{x'  +  H')  {x'  +  a')'  ~  \a'b  (a  +  bY  ' 

dx 


30.   Derive  an  integral  by  integrating 


7t 

o  «"  +  x^       2a 


tan-  '  ^  —  tan-  '  -^     —  =  -  log  =^  . 
.V  a.-J  ,v        2         g 


31.  Derive  a  definite  integral  by  integrating 


e  -  '"X  sin  ;?ji;  rti'j; 


with  reference  to  n. 


■  (cos  ax  —  cos  /7jc)  dx  =  —  log 


2         m    +  a 


32.  Derive  a  definite  integral  from  the  integral  employed  in  Ex.  31 
by  integration  with  reference  to  ;;/. 


s-"-^  —  s-^"^  \  dx  ~ 

Jo      ^-      L  J  L 


tan  ~  '  -   —  tan 
n 


ll- 


122 


METHODS  OF  INTEGRATION.  [Ex.  VII. 


33.   Derive  an  integral  by  integrating  with  respect  to  »i 


m 


£-ti!x  (,Q5  fix  dx  =:    — r- 


e-n.r  _  e-ix  I  b'   +  fi" 

COS  nx  dx  =  —  log  -5 ; 


1        34.  Derive  an  integral  by  integrating  with  respect  to  n  the  integral 
used  in  the  preceding  example. 

rE-"'^  ,  .  .     ,   N   ,  Ma  -  b) 

(sm  ax  —  sin  bx)  dx  =  tan" '  — , r  . 

Jo     A'  m    +  ab 

35.  Show  by  means  of  the  result  of  Ex.  32  that 

1: 


sin  nx  ,  TT 

dx  =  —  . 

X  2 


36.  Derive  an  integral  by  integration  from  the  result  of  Ex.  II.,  67, 


1> 


tan 


-•^-tan-'-^  I    r:   ..  =  -^  log 


37.  Evaluate       log  -^ w^^  by  the  method  of  Art.  96. 

Jo           .V    +  ^ 

7r{a-  b). 

38,  Evaluate       log 
•'  0 

I  +  — 3     \ogxdx.                      7ta  (log  a  —  i). 

§  VilL]  PLANE   AREAS.  1 23 

CHAPTER    III. 

Geometrical  Applications. 

VIII. 

Plane   Areas. 

102.  The  first  step  in  making  an  application  of  the  Inte- 
gral Calculus  is  to  express  the  required  magnitude  in  the  form 
of  an  integral.  In  the  geometrical  applications,  the  magni- 
tude is  regarded  as  generated  while  some  selected  independ- 
ent variable  undergoes  a  given  change  of  value.  The  inde- 
pendent variable  is  usually  a  straight  line  or  an  angle,  varying 
between  known  limits ;  the  required  magnitude  is  either  a 
line  regarded  as  generated  by  the  motion  of  a  point,  an  area 
generated  by  the  motion  of  a  line,  or  a  solid  generated  by  the 
motion  of  an  area.  A  plane  area  may  be  generated  by  the 
motion  of  a  straight  line,  generally  of  variable  length,  the 
method  selected  depending  upon  the  mode  in  which  the 
boundaries  of  the  area  are  defined. 


An  Area  Generated  by  a  Variable  Line  having  a  Fixed 

Direction. 

103.  The  differential  of  the  area  generated  by  the  ordinate 
of  a  curve,  whose  equation  is  given  in  rectangular  coordinates, 
has  been  derived  in  Art.  3.  The  same  method  may  be  em- 
ployed in  the  case  of  any  area  generated  by  a  straight  line 
whose  direction  is  invariable. 


124 


GEOMETRICAL  APPLICA  TIONS. 


[Art.  103. 


Fig. 


Let  AB  be  the  gencratiiii^  line,  and  let  R  be  its  intersection 
with  a  fixed  line  CD,  to  which  it  is  always 
perpendicular.  Suppose  R  to  move  uni- 
formly along  CD,  and  let  RS  be  the  space 
described  by  R  in  the  interval  of  time,  dt. 
Then  the  value  of  the  differential  of  the 
area,  at  the  instant  when  the  generating  line 
passes  the  position  AB,  is  the  area  which 
would  be  generated  in  the  time  dt,  if  the 
rate  of  the  area  were  constant.  This  rate 
would  evidently  become  constant  if  the  generating  line  were 
made  constant  in  length  ;  and  therefore  the  differential  is  the 
rectangle,  represented  in  the  figure,  whose  base  and  altitude 
are  AB  and  RS ;  that  is.  it  is  tJic  product  of  the  generating  line, 
and  the  differential  of  its  motion  in  a  direction  perpendicular  to 
its  length. 

104.  In  the  algebraic  expression  of  this  principle,  the  inde- 
pendent variable  is  the  distance  of  R  from  some  fixed  origin 
upon  CD,  and  the  length  of  AB  is  to  be  expressed  in  terms 
of  this  independent  variable. 

When  the  curve  or  curves  defining  the  length  oi  AB  are 
given  in  rectangular  coordinates,  CD  is  generally  one  of  the 
axes;  thus,  if  the  generating  line  is  the  ordinate  of  a  curve, 
the  differential  \s  y  dx,  as  shown  in  Art.  3.  It  is  often,  how- 
ever, convenient  to  regard  the  area  as  generated  by  some 
other  line. 

■  For  example,  given  the  curve  known  as  the  witch,  whose 
equation  is 

y  X  —  2a j^  +  4/7lv  =  0 (i) 


This  curve  passes  through  the  origin,  is  symmetrical  to  the 
axis  of  X,  and  has  the  line  .v  =  2a  for  an  asymptote,  since 
X  =  2a  makes  y  =  ±  00  . 

Let  the  area  between  the  curve  and  its  asymptote  be  re- 


§  VIII.]    AREAS  GENERATED  BY    VARIABLE  LINES. 


125 


quired.     We  may  regard   this  area  as  generated  by  the  line 
PQ  parallel  to  the  axis  of  x,  y  being  taken 
as  the  independent  variable.     Now 


PQ  =  2a  —  Xy 
hence  the  required  area  is 


A 


=        {2a  —  x)  dy  .     .     . 

J    -  CO 


(2) 


From  the  equation  (i)   of  the  curve,  we 
have 


whence 


2a  —  X  = 


y  +  4«' 


,2' 


and  equation  (2)  becomes 
dy 


Fig.  4. 


A  =  8«3 


-  ,-4«nan-'^ 

_„/+4«-  2a 


=  47ra- 


Oblique  Coordinates. 

105.  When  the  coordinate  axes  are  oblique,  if  a  denotes 
the  angle  between  them,  and  the  ordinate  is  the  generating 
line,  the  differential  of  its  motion  in  a  direction  perpendicular 
to  its  length  is  evidently  sin  a-dx ;  therefore,  the  expression 
for  the  area  is 


A  =  sin  a 


\y  dx. 


126  GEOMETRICAL  APPLICATIONS.  [Art.  105. 

As  an  illustration  let  the  area  between  a  parabola  and  a  chord 
passing  through  the  focus  be  required.  It  is  shown  in  treatises 
on  conic  sections,  the  expression  for  a  focal  chord  is 

yi^  =  4rtCosec^rt',      .      .     .      (i) 

where  a  is  the  inclination  of  the  chord 
to  the  axis  of  the  curve,  and  a  is  the 
distance  from  the  focus  to  the  vertex. 
It  is  also  shown  that  the  equation  of 
the  curve  referred  to  the  diameter 
which  bisects  the  chord,  and  the  tan- 
gent at  its  extremity  which  is  parallel  to  the  chord  is 

y  =  4^  cosec^  ci'X (2) 

The  required  area  may  be  generated  by  the  double  ordi- 
nate in  this  equation;  and  since  from  (i)  the  final  value  of 
y  \s  -^  2a  cosec^  a,  equation  (2)  gives  for  the  final  value  of  x 

OR  =  a  cosec^  a. 
Hence  we  have 

ta  cosec'a 

^  =  2  sin  n  y  dx, 

J  o 

or  by  equation  (2) 

'  <=°^e<='"  ,        2>a^  cosec^  a 


ta  cosec'a 

A  =  4Va  \  \/x  dx  = 

J  o 


Employment  of  an  Auxiliary    Variable. 
106.  We  have  hitherto  assumed  that,  in  the  expression 

A^^ydx, 


§  VI 1 1 .]     EMPLO  YMENT  OF  AN  A  UXILIAR  V  VARIABLE.        1 27 

X  is  taken  as  the  independent  variable,  so  that  dx  may  be 
assumed  constant ;  and  it  is  usual  to  take  the  limits  in  such  a 
manner  that  dx  is  positive.  The  resulting  value  of  A  will 
then  have  the  sign  of  y,  and  will  change  sign  if  y  changes 
sign. 

It  is  frequently  desirable,  however,  as  in  the  illustration 
given  below,  to  express  both  y  and  dx  in  terms  of  some  other 
variable.  When  this  is  done,  it  is  to  be  noticed  that  it  is  not 
necessary  that  dx  should  retain  the  same  sign  throughout  the 
entire  integral.  The  limits  may  often  be  so  taken  that  the  ex- 
tremityof  the  generating  ordinate  must  pass  completely  around 
a  closed  curve,  and  in  that  case  it  is  easily  seen  that  the  com- 
plete integral,  which  represents  the  algebraic  sum  of  the  areas 
generated  positively  and  negatively,  will  be  the  whole  area  of 
the  closed  curve. 

(07.  As  an  illustration,  let  the  whole  area  of  the  closed 
curve 


-a)    +  \b 


=  I, 


represented  in  Fig.  6,  be  required.     If  in  this  equation  we  put 


s.n  ^', 


we  shall  have 


^j    =cos?/?; 


whence 


x  =  a  sin^  -■/',  and  y  =  b  cos^ '/? . 


(I) 


Therefore 


y  dx  —  '^ab 


V 


cos^  '/.'  sin^  rp  dip. 


128 


GEOMETRICAL  APPLICATIONS. 


[Art.  107. 


Now  if  in  this  integral  we  use  the  limits  o  and  2;r,  the  point 

determined  by  equation  (i)  de- 
scribes the  whole  curve  in  the 
direction  A  BCD  A.  Hence  we 
have  for  the  whole  area 


A 


=  ^ad     cos*  tp  sin^  '/•  dtl', 
and  by  formula  (Q) 


A  =  lab  i 2/T  =  ^— — . 

The  areas  in  this  case  are  all  generated  with  the  positive 
sign,  since  when  y  is  negative  dx  is  also  negative.  Had  the 
generating  point  moved  about  the  curve  in  the  opposite  direc- 
tion, the  result  would  have  been  negative. 


Area  generated  by  a  Rotating  Line  or  Radius  Vector. 

108.  The  radius  vector  of  a  curve  given  in  polar  coordinates  is 
a  variable  line  rotating  about  a  fixed  extremity.     The  angular 

rate  is  denoted  by  —    and   may   be 


dt 


re- 


garded as  constant,  although  the  rate  at 
which  area  is  generated  by  the  radius 
vector  OP,  Fig.  7,  is  not  constant,  be- 
cause the  length  of  OP  is  not  constant. 
The  differential  of  this  area  is  the 
area  which  would  be  generated  in  the 
time  dt,  if  the  rate  of  the  area  were  con- 
stant ;    that  is  to  say,  if   the    radius  vector  were  of   constant 


Fig. 


§  VIII.]       AREAS  GENERATED  BY  ROTATING  LINES.  I29 


length.     It  is  therefore  the  circular  sector  O PR  oi  which,  the 
radius  is  r  and  the  angle  at  the  centre  is  dd. 


Since 


arc  PR^r  dd, 


sector  OPR  =  -r^  dd; 


therefore  the  expression  for  the  generated  area  is 


A 


r"  dd 


109.  As  an  illustration,  let  us 
find  the  area  of  the  right-hand  loop 
of  the  lemniscata 

7^  ^  o?  cos  2d. 


Fig.  8. 


(I) 


The  limits  to  be  employed  are   those    values  of    d  which 

make  r  =  o ;  that  is and  -, 

4  4 

Hence  the  area  of  the  loop  is 


A=—l'    cos  2^  ^6*=: -sin  28 
2]^^  4 


110.  When  the  radii  vectores,  r«  and  ri  corresponding  to  the 
same  value  of  6  in  two  curves,  have  the  same  sign,  the  area 
generated  by  their  difference  is  the  difference  of  the  polar  areas 
generated  by  7\  and  r^.     Kence  the  expression  for  this  area  is 


A=~\{ri-r,') 


dd. 


(2) 


130 


GEOMETRICAL  APPLICATIONS. 


[Art.  III. 


III.  Let    us    apply  this  formula    to    find    the   whole    area 
between  the  cissoid 

rj  =  2a  (sec  B  —  cos  6), 

Fig.  9,    and    its    asymptote   BP<i,  whose 
polar  equation  is 

ro  =  2a  sec  B. 

One  half  of  the  required  area  is  generated 
by  the  line  PxP-iy  while  6  varies  from  o  to 

—  n.     Hence  by  the  formula 


A  =2a^l'  (2-cos26') 


./e 


Fig.  g. 


Therefore  the  whole  area  required  is  S^^- 


Transformation  of  the  Polar  Formulas. 

112.  In  the  case  of  curves  given  in  rectangular  coordinates, 
it  is  sometimes  convenient  to  regard  an  area  as  generated  by  a 
radius  vector,  and  to  use  the  transformations  deduced  below 
in  place  of  the  polar  formulas. 


Put 


y  =  inx ; 


(I) 


now  taking  the  origin  as  pole  and  the  initial  line  as  the  axis 
of  X,  we  have 

X  ■=  r  cos  6,  y  ^=  r  sin  6  ;     .     .     .     (2) 


therefore 
and 


in  =  -  =  tan  6, 

X 


dm  =  sec"  e  dd (3) 


§  VIII.]     TRANSFORMATION  OF  THE  POLAR  FORMULAS.      I3I 
From  equations  (2)  and  (3), 

;f  ^  dm  =  r^  dd  ', 

therefore  equation  (i)  of  Art.  108  gives 


2 


x^  dm.      ......      (4) 


In  like  manner,  equation  (2)  Art.  no  becomes 


A=- 


2 . 


{x^-xl)dm (5) 


113.  As  an  illustration,  let  us  take  the  folium 

:!^^f—iaxy~o (i) 

Putting  J)/  =  vix,  we  have 

;r^  ( I  -f  v^)  —  lam^P'  =0 (2) 

This  equation  gives  three  roots  or  values  of  x^  of  which  two 
are  always  equal  zero,  and  the  third  is 


~  I  +  m^ ' 


(4) 


whence 


y 


'X^amr 

I   +  v^ 


(5) 


These  are  therefore  the  coordinates  of  the  point  P\\\  Fig.  lO. 
Since  the  values  of  x  and  y  vanish  when  ;;/  =  o,  and  when 
w  =  00 ,  the  curve  has  a  loop  in  the  first  quadrant.     To  find 


132 


GE  OME  TRICA  L  A  r PLICA  TIOXS. 


[Art.   113. 


the  area  of  this  loop  we  therefore  have,  by  equation  (4)  of  the 
preceding  article, 


_.9^  r   n^dm     _  _  ^a^      i      "I"  ^  3^2 
~~   2    Jo(i  +  m^f~        2    I  +  in^A^  ~    2 

114.  The  area  included  between  this  curve  and  its  asymp- 
tote may  be  found  by  means  of  equation 
(5),  Art.  112.  The  equation  of  a  straight 
line  is  of  the  form 


y  =  Dix  +  b, 


D\     0 
C 


Fig.  10. 


and  since  this  line  is  parallel  to  y  —  vix, 
the  value  of  in  for  the  asymptote  must  be 

that  which  makes  x  and  y  in  equations  (4)  and  (5)  infinite ; 

that  is,  ni  —  —  I  ;  hence  the  equation  of  the  asymptote  is 


y  +  X  —  b, 


(6) 


in  which  h  is  to  be  determined.  Since  when  m  =  —  i,  the 
point  P  of  the  curve  approaches  indefinitely  near  to  the  asymp- 
tote, equation  (6)  must  be  satisfied  by  P  when  m—  —  i. 
From  (4)  and  (5)  we  derive 


y  -\-  X  =  ^a 


m^  +  7n 


lam 


I  +  n^       I  —  w  +  m^  ' 
whence,  putting  ju  —  —  i,  and  substituting  in  equation  (6) 

—  a  =  b, 
the  equation  of  the  asymptote  AB,  Fig.  10,  is 

J  +  'V  =  -  ^ (;) 


§  VIII.]     TRANSFORMATION  OF  THE  POLAR  FORMULAS'      133 

Now,  as  in  varies  from  —  oo  to  o,  the  difference  between  the 
radii  vectores  of  the  asymptote  and  curve  will  generate  the 
areas  OBC  zxii\  ODA,  hence  the  sum  of  these  areas  is  repre- 
sented by 


A  =  ~\      {xl  —  x^)  dm, 

2j_00 


in  which  x^  is  taken  from  the  equation  of  the  asymptote  (7), 
and  Xi  from  that  of  the  curve. 
Putting  y  =  jux,  in  (7),  we  have 


X2  — 


a 

I    +  JU 


and  the  value  of  x^  is  given  in  equation  (4).     Hence 


A  = 


<^nr' 


_(i  +  vif       (i  +  n?f  _ 


dm 


I  +  nf        I  +  m 


a^  2  ■\-  m  —  iir 
2        I  +  m^ 


a^       2  —  m 
2    I  ~  7n  +  m^ 


—  /,2 


Adding  the  triangle  OCD,  whose  area  is  la^,  we  have  for  the 
whole  area  required  |^^. 


*  This  reduction  is  given  to  show  that   the  integral  is  not  infinite  for  the 
value  m=:  —  i,  which  is  between  the  limits.     See  Art.  77. 


134 


GEO  ME  TRIG  A  L  APPLIGA  7  IONS.  [Ex.  V I  IT. 


'  Examples  VIII. 

1.  Find  the  area  included  between  the  curve 

d'y  =  ^3  ^  ax\ 
and  the  axis  of  x. 

2.  Find  the  whole  area  of  the  curve 

ay  =  X*  {a'  -  x\ 

3.  Find  the  area  of  a  loop  of  the  curve 

'V'(^'+y)=y(^'-/). 

4.  Find  the  area  between  the  axes  and  the  curve 

y  (.v'  +  d')  =b'{a-  x).  b' 

5.  Find  the  area  between  the  curve 


7ta 
4 


'■-(.-.). 


7t         log  2 


L4 


]• 


xy  +  ay   —ax  =.  o, 


and  one  of  its  asymptotes. 


2a. 


6.  Find  the  area  between  the  parabola y  =  ^ax  and  the  straight 

line  y  ^=  X.  — . 

3 

7.  Find  the  area  of  the  ellipse  whose  equation  is 

ax  +  2bxy  +  <-v   =  1. 


^iac-by 


§VIII.]  EXAMPLES.  135 

8.  Find  the  area  of  the  loop  of  the  curve 

cy^  ^  {x  —  d){x  —  by, 
m  which  c  >  o  and  0  >  a.  — ^^ ~-  • 


32^8 


9.  Find  the  area  of  the  loop  of  the  curve 

ay  =  x*{b  +  x). 

105^3 

10.  Find  the  area  included  between  the  axes  and  the  curve 


ft -{if-  t 


II.  Ti  n  is  an  integer,  prove  that  the  area  included  between  the 
axes  and  the  curve 


©""-(i) 


+  i^r  =  . 


.  n{n—  1) '  '  •  1  , 

IS  A  =  — - — ^ r-^ — ,  .  ab. 

211  [2/1  —  I)  •  '  '  [n  +  I) 


12.  If  nis  an  odd  integer,  prove  that  the  area  included  between 
the  axes  and  the  curve 


0)"-(l)'  = 


_  \fi{n—  2)"  •  ij  Ttab 
IS  ^  ^—        7  c  • 

2n  {2n  —  2}  •  •  •  2     2 


136 


GEOMETRICAL  AJPPLICATIONS.  [Ex-  VIII. 


13.  In  the  case  of  the  curtate  cycloid 

X  =  fl-'/'  —  b  sin  '/',  y  =  a  —  b  cos  ?/', 

find  the  area  between  the  axis  of  -v  and  the  arc  below  this  axis. 


(2a' +  Z'')  cos -'I  -  2,^  Vib' -  a'). 


14.  If  ^  =  ^art,  show   that  the   area   of  the  loop  of  the  curtate 
cycloid  is 

15.  Find  the  area  of  the  segment  of  the  hyperbola 

-r  =  a  sec  ip,  y  =  b  tan  ?/', 

cut  off  by  the  double  ordinate  whose  length  is  2b. 


ab 


V2  —  log  tan  — 


16.  Find  the  whole  area  of  the  curve 

r'  =  a''  cos'  0  +  b"  sin'  0. 

17.  Find  the  area  of  a  loop  of  the  curve 

r'  =  a'  cos"  0  —  ^'  sin'  0. 


-  (a'  +  b'). 

2 


ab       {a'-b')        .,a 

—  +  ^^- -'  tan      -. 

22  b 


18.  Find  the  areas  of  the  large  and  of  each  of  the  small  loops  of 
the  curve 

r  ■=  a  cos  0  cos  2O  ; 


§  VIII.]  EXAMPLES.  137 

and  show  that  the  sum  of  the  loops  may  be  expressed  by  a  single 
integral. 

na^   _  «*  .       TTa"      a" 

—T-  +  -  ,     and . 

16        4  '  32       8 

19.  In  the  case  of  the  spiral  of  Archimedes, 

r  =  aB, 

find  the  area  generated  by  the  radius  vector  of  the  first  whorl  and 
that  generated  by  the  difference  between  the  radii  vectores  of  the  nth. 
and  {u  +  i)th  whorl. 


6 

20.  Find  the  area  of  a  loop  of  the  curve 


and      Sna  n 


na 
12 


r  =  «  sm  Tfi, 

21.  Find  the  area  of  the  cardioid 

r  =  4rt!  sin'  4^0.  ^nc^. 

22.  Find  the  area  of  the  loop  of  the  curve 

cos  29  «"  (4  —  n\ 

r  ^=^  a .  - — ^-^^ -, 

cos  (5  2 

23.  In  the  case  of  the  hyperbolic  spiral, 

r^  =  a, 

show  that  the  area  generated  by  the  radius  vector  is  proportional  to 
the  difference  between  its  initial  and  its  final  value. 


138  GEOMETRICAL  APPLICATIONS.  [Ex.  VIII. 

24.   Find  the  area  of  a  loop  of  the  curve 


r  -=■  a  cos  ti  0. 


va 
4« 


25.  Find  the  area  of  a  loop  of  tlie  curve 


3         .,  sin  -5G  a 

cos    i  2 


26.  Find  the  area  of  a  loop  of  the  curve 


r  sin  0  =  (7  cos  2'^ 


Notice  that  r  /V  r^a/  and  finite  from  0  =  ^—  to^i  =:  —  ,  and  that    -. — 
•^         -^  4  4  '  J  sm  G 

iV  negative  in  this  interval.  ^M    1^2  —  log  (i  +  Vs)     . 

27.  Find  the  area  of  a  loO])  of  the  curve 

(x-  +fr^a\xy, 

a' 
Transform  to  polar  coordinates.  — . 

28.  In  the  case  of  the  lima9on 

;'  =  2a  cos  6r  +  <5, 

find  the  whole  area  of  the  curve  when  /'  >  2a  and  show  that  the  same 
expression  gives  the  sum  of  the  loops  when  b  <  2a, 

{20"  +  lr)7t. 


139 


§  VIII.]  EXAMPLES. 

29.   Find  separately  the  areas  of  the  large  and  small  loops  of  the 
limacon  when  b  <.2a. 

\i  a  =  COS"'  (  —  — 

large  loop  =  {20"  +  b"")  a  +  ^  ^(4^^  -  I,') ; 
small  loop  =  {20^  +  b")  [tt  —  a)  —  —  ^(4^^  —  i>'). 


30.  Find  the  area  of  a  loop  of  the  curve 

r^  =  a^  cos  nfi  +  b'  sin  71  9. 

31.  Find  the  area  of  the  loop  of  the  curve 

2  cos  26  —  I 

r  =  a -7, , 

cos  u 


V(a'  +  b*^ 


'^'-T. 


32.   Show  that  the  sectorial  area  between  the  axis  of  -v,  the  equi- 
lateral hyperbola 


.r^-/=i, 


and  the  radius  vector  making  the  angle  ^  at  the  centre  is  represented 

by  the  formula 

.        I  ,       I  +  tan  0 

/I  =:  -  log ; 

4      ^  I  —  tan  fj ' 

and  hence  show  that 

X  =  ■ ,  and  y  =  . 


If  A  denotes  the  corresponding  area  m  the  case  of  the  circle 
we  have 


.r'+/=i, 


X  =  cos  2A,  and  y  =  sin  2 A. 


140  GEOMETRICAL  APrLICATIONS.  [Ex.  VII I. 

In  accordance  with  the  analogy  thus  presented^  the  values  of  x  and  y  given 
above  are  called  the  hyperbolic  cosine  and  the  hyperbolic  sine  of  2  A.     Thus 

£2A       I       J-2A  ^-2A  ^2A 

=■  cosh  (2^),  =  sinh  (2^), 


33.  Find  the  area  of  the  loop  of  the  curve 

A"*  —  sai^y  +  2ay^  =  o. 

34.  Find  the  area  of  the"  loop  of  the  curve 


38.  Trace  the  curve 


.    y 

x  —  2a  sin  — , 

-V 


35-^' 


^■2«  +  i  -f-y«  +  i  —  (2//  +  i)  a.vy.  a  . 

35.  Find  the  area  between  the  curve 

x^»  +  ^  jf.y^n+1  —  (2,/  4-  i)  ax"y" 

2n  +  I    J 
and  its  asymptote.  ^^ 


36.  Find  the  area  of  the  loop  of  the  curve 

y  4-  ax^  —  axy  =  o. 

37.  Find  the  area  of  a  loop  of  the  curve 

X*  +  y*  =  a^xy. 


a 
6^ 


■na 


and  find  the  area  of  one  loop.  nc^ 


§ix.] 


VOLUMES  OF  GEOMETRIC  SOLIDS. 


141 


IX. 


Volumes  of  Geo7iietric  Solids. 

115.  A  geometric  solid  whose  volume  is  required  is  fre- 
quently defined  in  such  a  way  that  the  area  of  the  plane  sec- 
tion parallel  to  a  fixed  plane  may  be  expressed  in  terms  of 
the  perpendicular  distance  of  the  section  from  the  fixed  plane. 
When  this  is  the  case,  the  solid  is  to  be  regarded  as  generated 
by  the  motion  of  the  plane  section,  and  its  differential,  when 
thus  considered,  is  readily  expressed. 

116.  For  example,  let  us  consider  the  solid  whose  surface  is 
formed  by  the  revolution  of  the  curve  APB,  Fig.  11,  about 
the  axis  OX.  The  plane  section  per- 
pendicular to  the  axis  OX  \s  a  circle; 
and  if  APB  be  referred  to  rectangu- 
lar coordinates,  the  distance  of  the 
section  from  a  parallel  plane  passing 
through  the  origin  is  x,  while  the 
radius  of  the  circle  \s  y.  Supposing 
the  centre  of  the  section  to  move 
uniformly  along  the  axis,  the  rate  at 
which  the  volume  is  generated  is  not 
uniform,  but  its  differential  is  the  vol- 
ume which  would  be  generated  while  the  centre  is  describing 
the  distance  dx,  if  the  rate  were  made  constant.  This  differen- 
tial volume  is  therefore  the  cylinder  whose  altitude  is  dx,  and 
the  radius  of  whose  base  is  j.     Hence,  if  F  denote  the  volume, 

dV  =  7ty^  dx. 

117.  As  an  illustration,  let  it  be  required  to  find  the  volume 
of  the  paraboloid,  whose  height  is  h,  and  the  radius  of  whose 
base  is  b. 


142  GEOMETRICAL  APPLICATIONS.  [Art.  1 1 7. 

The  revolving  curve  is  in  this  case  a  parabola,  whose  equa- 
tion is  of  the  form 

and  since  y  =  b  when  x  =  //, 

d^  =  4a/i,  whence  4rz  =  y  ; 


/i 


the  equation  of  the  parabola  is  therefore 

Hence  the  volume  required  is 
y=  Tt  \    y"  dx 


71  — -       X  dx  — . 

//Jo  2 


118.  It  can  obviously  be  shown,  by  the  method  used  in 
Art.  116,  that  whatever  be  the  shape  of  the  section  parallel  to 
a  fixed  plane,  the  differential  of  the  volume  is  the  product  of  the 
area  of  the  generatitig  section  and  the  differential  of  its  motion 
perpendicular  to  its  pla?ie. 

If  the  volume  is  completely  enclosed  by  a  surface  whose 
equation  is  given  in  the  rectangular  coordinates  .r,  j,  s,  and  if 
we  denote  the  areas  of  the  sections  perpendicular  to  the  axes 
by  Ajcy  Ay,  and  A^,  we  may  employ  either  of  the  formulas 

V=  [A^dx,  V=  lAydv,  V=  [A.ds. 


The  equation  of  the  section  perpendicular  to  the  axis  of  x 
is  determined  by  regarding  x  as  constant  in  the  equation  of 
the  surface,  and  its  area  A^  is  of  course  a  function  of  x. 


§  IX.]  VOLUMES  OF  GEOMETRIC  SOLIDS.  1 43 

For  example,  the  equation  of  the  surface  of  an  ellipsoid  is 

■ h  —  +  —  =  I 

The  section  perpendicular  to  the  axis  of  x  is  the  ellipse 

jj/2         ^  ^  ^2  _  ,j,2 

^  "^  ?  ~cF' '  . 


b  c 

whose  semi-axes  are-   ^/ic?  —  x^)  and  -  V(a^  —  x^. 
a      ^  a     ^  ' 

Since  the  area  of  an  eUipse  is  the  product  of  n  and  its  semi- 
axes, 

a''  ' 

The  limits  for  x  are  ±a,  the  values  between  which  x  must  lie 
to  make  the  ellipse  possible.     Hence 

a^  i-a  3 

(19.  The  area  A,^  can  frequently  be  determined  by  the  con- 
ditions of  the  problem  without  finding  the  equation  of  the 
surface.  For  example,  let  it  be  required  to  find  the  volume  of 
the  solid  generated  by  so  moving  an  ellipse  with  constant 
major  axis,  that  its  center  shall  describe  the  major  axis  of  a 
fixed  ellipse,  to  whose  plane  it  is  perpendicular,  while  the  ex- 
tremities of  its  minor  axis  describe  the  fixed  eUipse.  Let  the 
equation  of  the  fixed  ellipse  be 

—   -1-  —  =.  I 
^2  ^  b^        ' 


144 


GEOMETRICAL  APPLICATIOXS. 


[Art.  I  19. 


and  let  c  be  the  major  semi-axis  of  the  moving  ellipse.  The 
minor  semi-axis  of  this  ellipse  is  y.  Since  the  area  of  an 
ellipse  is  equal  to  n  multiplied  by  the  product  of  its  semi-axes, 
we  have 


a 


Ttbc  (" 
Therefore  V= —         V{a^  —  -r^)(^-v; 

U      J  —a 

hence,  see  formula  (J/), 


V  = 


i^abc 


The  Solid  of  Revoliltio7i  regarded  as  Generated  by  a 
Cylindrical  Stirface. 

120.  A  solid  of  revolution  may  be  generated   in  another 
ji  manner,  which  is  sometimes  more 

convenient  than  the  employment 
of  a  circular  section,  as  in  Art.  1 16. 
For  example,  let  the  cissoid  FOR, 
Fig.  12,  whose  equation  is 

..."•"■■^  y^  {2a  —  x)  =  x^, 

revolve  about  its  asymptote  AB. 
The  line  PR,  parallel  to  AB  and 
terminated  by  the  curve,  describes 
a  cylindrical  surface.  If  we  con- 
ceive the  radius  of  this  cylinder  to 
pass  from  the  value  GA  =  2a  to  zero,  the  cylindrical  surface 
will  evidently  generate  the  solid  of  revolution.      Now  every 


Ji. 

1 

\ 

A — 

^ 

y 

.4 

\ 

T" 

p 

? 

/ 

Fig.  12. 


IX.] 


DOUBLE  INTEGRATION. 


145 


point  of  this  cylindrical  surface  moves  with  a  rate  equal  to 
that  of  the  radius;  therefore  the  differential  of  the  solid  is 
the  product  of  the  cylindrical  surface,  and  the  differential  of 
the  radius.     The  radius  and  altitude  in  this  case  are 

PC=  ia  —  X,  and  PR  =  2j, 

therefore  V  =  47T  \    {2ax  —  x^)-x  dx. 

Putting  X  —  a  =  a  sin  6, 

V=  47ra^    '  „(cos^  6  +  cos^  6  sin  6)  dB  —  27rV. 
2 

Do2iblc   l7itco;7'ation. 

121.  When  rectangular  coordinates  are  used,  the  expression 
for  the  area  generated  by  a  line  parallel 
to  the  axis  of  y  and  terminated  by  two 
curves  is 


^  =       (ja-Ji)^-^- 


(I) 


A  C 

Fig.  13. 


Let  AB,  in  Fig.   13,  be  the  initial, 
and  CD  the  final  position  of  the  gen- 
erating line,  then  the  area  is  ABDC,  which  is  enclosed  by  the 
curves 

and  by  the  straight  lines 


X  =  b. 


146  GEOMETRICAL  APPLICATIONS.  [Art.  121. 

If  in  equation  (i)  we  substitute  for  jj  —  Ji  the  equivalent  ex- 
pression    dy,  we  have 

A=\'  \''dydx, (2) 

which  expresses  the  area  in  the  form  of  a  double  integral.  In 
this  double  integral  the  limits  j'l  and  _;'2  forj',  are  functions  of 
x,  while  a  and  d,  the  limits  for  x,  are  constants. 

122.  If  the  area  is  that  of  a  closed  curve  j'l  and  j'2  are  two 
values  of  J  corresponding  to  the  same  value  of  x  in  the  equa- 
tion of  the  curve,  and  a  and  d  are  the  values  of  x  for  which  j'j 
and  j2  become  equal,  as  represented  by  the  dotted  lines  in  Fig. 
13.  It  is  evident  that  the  entire  area  may  also  be  expressed  in 
the  form 

A  =  ^yxdy; (3) 

and  that  when  either  of  the  forms  (2)  or  (3)  is  applied  to  the 
area  of  a  closed  curve  the  limits  are  completely  determined  by 
the  equation  of  the  curve. 

123.  The  limits  in  either  of  the  expressions  (i)  or  (2)  define 
a  certain  closed  boundary,  and  since  either  of  these  integrals 
represents  the  included  area,  it  is  evident  that  wc  may  write 


dj'  dx  =     dx  dy ; 


provided  it  is  understood  that  the  limits  in  the  two  expressions 
are  such  as  to  represent  the  same  boundary.  It  should  however 
be  noticed  that  if  the  boundary  is  like  that  represented  by  the 
full  lines  in  Fig.  13,  or  if  the  arcs  y  —  y\  and  y  =yi  do  not 
belong  to  the  sainc  curve,  we  cannot  make  a  practical  application 
of  the  form  (3)  without  breaking  up  t'he  integral  into  several 
parts. 


IX.] 


DOUBLE  INTEGRATION. 


147 


124,   Let  4>  {p-'iy)  t>e  any  function  of  x  and  j/.    In  the  double 
integral 

I     f  '  <}){x,y)dydx, (i) 


X  is  considered  as  a  constant  or  independent  of  y  in  the  first 
integration,  but  the  limits  of  this  integration  are  functions  of  x. 
The  double  integration  is  then  said  to  extend  over  the  area 
which  is  represented  by  the  expression 


\' dydx,        or         [    {y^-yi)dx. 

J  a    J.i ,  J  a 


(2) 


125.  Now  let  the  surface,  of  which 
S~  (I)  (.r,  y)      . 


(3) 


is  the  equation  in  rectangular  coordinates,  be  constructed ;  and 
let  a  cylindrical  surface  be  formed  by  moving  a  line  perpen- 
dicular to  the  plane  of  xj  about  the  boundary  of  the  area  (2) 
over  which  the  integration  extends.  Let  us  suppose  the  value 
of  s  to  be  positive  for  all  values  of  x  and  y  which  represent 
points  within  this  boundary.  Then  the  cylindrical  surface, 
together  with  the  plane  of  xy  and  the  surface  (3),  encloses  a 
solid,  of  which  the  base  is  the  area 
(2)  in  the  plane  xy,  or  ASBR  in  Fig. 
14,  and  the  upper  surface  is  CQDP  a 
portion  of  the  surface  (3). 

Let  SRPQ  be  a  section  of  this 
solid  perpendicular  to  the  axis  of  x. 
In  this  section  x  has  a  constant  value, 
and  the  ordinates  of  R  and  5  are  the 
corresponding    values   of   j'l    and  y^.  ^ 

The  area  of  this  section,  which  denote  Fig.  14. 


P 

D 

c 

^-^ 

\y.. 

.■■■'" 

h 

X 

A 

L 

148  GEOMETRICAL   APPLICATIOXS.  [Art.   1 25. 

by  A.ry  as  in  Art.  1 1  7,  may  be  regarded  as  generated  by  the  line 
z,  hence 


hi 


and  therefore 


V=  I      P%  rt^;/ rt'.v, (l) 

which  is  identical  with  expression  (i)  Art.  124. 

126-  Now   it  is  evident  that  the  same  volume  may  be  ex- 
pressed by 


V=  \\::dxdy, 

provided  that  the  double  integration  exte?ids  over  the  same  area. 
Hence,  with  this  understanding,  we  may  write 

J  U  (-^' jO  dydx  =  \\<f>  {x,  y)  dx  dy. 

In  this  formula  x  and  y  may  be  regarded  as  taking  the 
places  of  any  two  variables,  the  limits  of  integration  being 
determined  by  a  given  relation  between  the  variables.  Thus 
we  may  write 

(j)  {li,  t')  dv  du  —      (}>  (u,  v)  du  dv, 

provided  the  limits  of  integration  are  determined  in  each  case 
by  the  same  relation  between  u  and  v. 

127.   For  example,  if  this  relation  is 

u^  +  1^  —  e'^  =  o, 


§  IX.]  DOUBLE  INTEGRATION.  1 49 

the  range  of  values  in  the  first  integration  is  between 

that  is,  we  must  have 

v^  <  c^  —  u^, 

or  21^  +  v'^  —  c'^  <  o (i) 

But  this  condition  also  expresses  the  limits  for  ti,  since  v  is 
only  possible  when  t^  <  c^.  Now,  putting  rectangular  coordi- 
nates, X  and  J,  in  place  of  ti  and  v,  it  is  convenient  to  express 
the  restriction  (i),  by  saying  that  the  range  of  values  of  x  and 
y  is  such  as  to  represent  every  point  zvit/mi  the  circle 

;^  +  y  _  ^2  ^  o. 

Volumes  by  Dottble  and  Triple  Integration. 

128.  As  an  application  of  formula  (i),  Art.  125,  let  us  sup- 
pose the  curve  ASBR  to  be  the  circle 

{x-hf^-{v-kf  =  c\ (I) 

and  the  equation  of  the  surface  CQDP  to  be 

xy  =  pz (2) 

Then  V  = -\         xy dy  dx  = —  \    {y^-yl)xdx, 

p  Ja  iy^  2p  J  a 


150 


GEOMETRICAL   APPLTCATIO.VS.  [Art.  128. 


in  which  the  limits  j\  and  j,  are   derived   from    equation   (i). 
Hence 


and 


V=~  f "  Vic"  -  {x  -  hf]  X dx. 

P   ]a 


The  limits  for  ;ir  are  the  extreme  values  of  x  which  makej» 
possible  ;  that  is, 


a  =^  h  —  c 


and 


b  =  h  -V  c. 


To  evaluate  the  integral,  put 

X  —  h  =  c  sin  B ; 


then  V^—[\cos^d{h  +  c  sin  d)  dd. 

Since,  by  Art.  87, 

It 
V  ^zo^Bsm  Odd  =0, 


we  have  finally 


r= 


rkhc^ 


129.  A  volume  in  general  may  be  represented  by  the  triple 
integral 


F  = 


dz  dy  dx, 


(I) 


§  IX.]  TRIPLE  INTEGRATION.  15 1 

which  is  equivalent  to 

V=\\{z^-z^dydx^ (2) 

for     (^2  ~  -i)  (^  =  Ax^  the  area  of  a  section  perpendicular  to 

the  axis  of  x.  We  may  regard  this  formula  as  expressing  the 
difference  between  two  cylindrical  solids  of  the  form  represented 
in  Fig.  14. 

130.  When  the  volume  is  that  of  a  closed  surface,  z^  and  z-^ 
are  two  values  of  z  in  terms  of  x  and  y  found  from  the  equa- 
tion of  the  surface.  The  area  over  which  the  integration 
extends  is  in  this  case  the  projection  of  the  solid  upon  the  plane 
of  xy ;  in  other  words,  the  base  of  a  circumscribing  cylinder. 
Thus,  if  the  volume  is  that  of  the  sphere 

x^+  f  ^  {z-~cf=a^, (I) 

Zx  and  ^"2  are  the  two  values  of  z  derived  from  this  equation  • 
that  is  c±  V(n-  —  x^—y^). 

Hence  z^  —  Zi  =  2  V{a^  —  x^  —  f). 


and 


F=  2  [[  \'  {a"-  -  x'-f)  dydx.     c     .     .     .  (2) 


The  integration  here  extends  over  the  circle 

x'+f-a^^Q (3) 


152  GEOMETRICAL  APPLICATIONS.  [Art.   130. 

since  -Co  —  c^  is  real  only  when 

a^  —  x^  —  }'^  y  o. 

From  equation  (3)  we  find  the  limits  forjj/  to  be 

hence,  by  formula  {M),  equation  (2)  becomes 


V=  TlUd^  -x^)dx. 


Finally  the  limits  for  x  are  ±  a,  since  y  is  real  only  when  x  is 
between  these  limits  ; 


therefore  V 


r       I    1'' 

=   7t       C?X X^  = 

L  3      J -a 


^;r^. 


Elements  of  Area  and  Volume. 
131.  In  accordance  with  Art.  100,  the  expression  for  an  area, 

f    \\iydx (i) 


is  the  limit  of  the  sum 


Since  each  of  the  terms  included  in  2^''  Aj  is  multiplied  by 
the  common  factor  A.r,  this  sum  may  be  written  in  the  form 

:^l:^j,>jA;p (2) 


§  IX.]  ELEMENTS  OF  AREA   AND    VOLUME.  1 53 

The  sum  (2)  consists  of  terms  of  the  form 

and  this  product  is  called  tJic  clement  of  the  sum  ;  in  like  man- 
ner, the  product 

dy  dx, 

which  takes  the  place  of  Ay  Ax  when  we  pass  to  the  limit  by 
substituting  integration  for  summation,  is  called  the  element  of 
the  integral  (i),  or  of  the  area  represented  by  it. 

132.  We  may  now  regard  the  process  of  double  integration 
as  a  process  of  double  summation,  as  indicated  by  expression 
(2),  followed  by  the  act  of  passing  to  the  limiting  value.  In 
the  first  summation  indicated,  the  elemental  rectangles  corre- 
sponding to  the  same  value  of  x  are  combined  into  the  term 
(j2  —  J'l)  -^-^'j  which  may  be  called  a  linear  element  of  area,  since 
its  length  is  independent  of  the  symbol  A . 

133.  It  is  easy  to  see  that,  in  a  similar  manner,  when  rec- 
tangular coordinates  are  used,  a  volume  may  be  regarded  as 
the  limiting  value  of  the  sum  of  terms  of  the  form 

A X  Ay  A2\ 

and  hence  dx  dy  ds, 

which  takes  its  place  when  we  pass  to  the  limiting  value  by 
substituting  integration  for  summation,  is  called  tJie  element  of 
volume. 

If  the  summation  is  effected  in  the  order  z,  y,  x,  the  first 
operation  combines  the  elements  which  have  common  values 
oi  y  and  x  into  the  linear  element  of  volume^ 

(^2-  ^1)  AX  Ay. 


154  GEOMETRICAL   APPLICATIONS.  [Art.   I33, 

The  second  operation  combines  the  linear  elements  correspond- 
ing to  a  common  value  of  x,  over  a  certain  range  of  values  oi  y, 
into  a  term  whose  limiting  value  takes  the  form 

This  last  expression  represents  a  lainina  perpendicular  to  the 
axis  of  X,  whose  area  is  A^  a  section  of  the  solid,  and  whose 
thickness  is  a. v. 


Polar  Elements. 
134.  If  in  the  formula  for  a  polar  area, 

=  l|(;V-;f)^^, (I) 


A 


[equation  (2),  Art.  no],  we  substitute  for  -{r^—  r^)  the  equi\ 


alent  expression 


r  dr,  we  obtain 

\drdff, (2) 


in  which  (x  and  ft  are  fixed  limits  for  6. 

Now  it  follows,  from  Art.  126,  that  the  limits  being  deter- 
mined by  a  certain  relation  between  r  and  6,  this  integral  may 
also  be  put  in  the  form 

A=\\\''de.dr=\\{H^-e,)dr,     ...      (3) 


§  IX.]  POLAR  ELEMENTS.  1 55 

in  which  a  and  b  are  the  limiting  values  of  r,  between  which  6 
is  possible. 

The  expression  r  dr  dd, 

in  equation  (2),  is  called  t\\Q  polar  element  of  area.* 
135.  The  formula 


A^\^r{e^-e,)dr 


may  also  be  derived  geometrically ;  for  ;•  (^2  —  B^  is  the  length 
of  an  arc  whose  radius  is  r.  As  r  increases,  this  arc  generates 
the  surface,  and  it  is  plain  that  every  point  has  a  motion, 
whose  differential  is  dr,  in  a  direction  perpendicular  to  the  arc. 
136.  In  determining  the  volume  of  a  solid,  it  is  sometimes 
convenient  to  express  ^  as  a  function  of  the  polar  coordinates 
of  its  projection  in  the  plane  of  xy.  In  this  case  we  employ 
the  linear  element  of  volume, 

(^2  -  -1)  r  dr  dO, 
corresponding  to  the  polar  element  of  area. 


*  It  is  easily  shown  that  the  area  induded  between  the  circles  whose  radii  are 
rand  ;■  +  Ar,  and  the  radii  whose  inclinations  to  the  initial  line  are  9  and  9  +  A9 
is 

(r  +  ^  A  r)  A  r  A  6. 

Since  r+  iAr  is  intermediate  between  r  and  r  +  Ar,  the  limiting  value  of  the 
sum,  of  which  this  is  the  element,  is,  by  Art.  99,  the  integral  of  the  element 

In  the  summation  corresponding  to  equation  (i),  the  elements  are  first  combined 
into  the  sectorial  element 

l(r-i  _  r^)  A6  ; 

while  in  the  summation  corresponding  to  equation  (3),  they  are  first  combined  into 
the  arc-shaped  element 

(r  +  4Ar)(«2  —  di)  Ar. 


156  GEOMETRICAL  APPLTCATIONS.  [Art.  1 36. 


As  an  illustration,  let  us  determine  the  volume  cut  from  a 
sphere  by  a  right  cylinder,  having  a  radius  of  the  sphere  for 
one  of  its  diameters.  Taking  the  centre  of  the  sphere  as 
the  origin,  the  diameter  of  the  cylinder  as  initial  line,  and  the 
axis  of  z  parallel  to  the  axis  of  the  cylinder,  we  have  for 
every  point  on  the  surface  of  the  sphere 

^  +  r''  =  a\ (I) 

where  a  is  the  radius  of  the  sphere.     Hence 


and  V=  2\['\a^~r^)Kdrdft  =  [\'-  -(«2_^^i 


dd. 


The  circular  base  passes  through  the  pole,  and  its  equation  is 

r  =  a  cos  0, (2) 

hence  the  limits  for  r  are  o  and  a  cos  6,  and  by  substitution  we 
obtain 


2/7^   f 

r=—    (I  -sin^^)^^. 


The  limits  for  B  are  ±  - ,  the  values  which  make  r  vanish 

2 

in  equation  (2) ;  but  it    is  to  be  noticed   that  the  expression 

(^  —  r^)8,  for  which  we  have  substituted  ci?  sin^  6,  is  always  posi- 
tive, whereas  sin^  d  is  negative  in  the  fourth  quadrant.  Hence 
the  value  of  V  is  double  the  value  of  the  integral  in  the  first 
quadrant ;  that  is, 

V=  —       (i  —  sm'  8)  dd  = . 

3   Jo  ^  3  9 


§IX.] 


POLAR  ELEMENTS. 


157 


If  a  second  cylinder  whose  diameter  is  the  opposite  radius  of 
the  sphere  be  constructed,  the  whole  volume  removed  from  the 


sphere  is 


,  and    the   portion    of   the   sphere    which 


remains  is ,  a  quantity  commensurable  with  the  cube  of 

the  diameter. 


Polar   Coordinates  in  Space. 


137.  A  point    in    space  may  be  determined   by  the  polar 
coordinates  p,  ^,  and  ^,  of  which  p   de- 
notes the    radius    vector    OP,    Fig.    15, 
4)  the  inclination  POR  of  /a  to  a  fixed 
plane  passing  through  the  pole,  and  d 
the   angle    ROA,  which  the   projection 
of   p    upon    this    plane    makes    with    a 
fixed  line   in   the   plane.     The  angles  (}) 
and  9  thus  correspond    to  the    latitude 
and  longitude  of  the  point  P  considered 
as  situated  upon  the  surface  of  a  sphere 
whose  radius  is  p.     The  radius  of  the 
circle  of  latitude  BP  is 


-2 


Fig.  15. 


PC  =  p  cos  ^. 

The  motions  of  P,  when  p,  (f>,  and  6  independently  vary,  are 
in  the  directions  of  the  radius  vector  OP  and  of  the  tangents  at 
Pto  the  arcs  PR  and  PB.  The  differentials  of  these  motions 
are  respectively 


dp, 


pd^,         and         p  cos  ^dO; 


158 


GEOMETRICAL  APPLICATIOXS.  [Art.   1 37. 


and  since  these  motions  are  mutually  rectangular,  the  element 
of  volume  is  their  product, 

f?  cos  ^  dp  d^  dd, 

and  V  =\\    (^  cos  <j)dpd(f)dd (i) 

138.  Performing  the  integration  with  respect  to  p,  the  for- 
mula becomes 

V=-\^^{p^-  Pl)  cos  (f>d(l>  do (2) 


When  the  radius  vector  lies  entirely  within  the  solid,  the  lower 
limit  px  must  be  taken  equal  zero,  and  we  may  write 

V  = -\\p'' cos  (j)  d(l>  dd (3) 

The  element  of  this  double  integral  has  the  form  of  a  pyramid 
with  vertex  at  the  pole. 

If,  on  the  other  hand,  in  formula  (i)  we  perform  first  the 
integration  with  respect  to  (f>,  we  have 


V  =      (sin  ^  —  sin  (f>^  p-  dp  dO. 


(4) 


Taking  the  lower  limit  <f)x  =  o,  so  that  the  solid  is  bounded  by 
the  plane  OR  A,  we  have  the  simpler  formula 

V=  {{sin  (^f?  dp  dd (5) 

139.  The   formulas  of    the  preceding  article  take  simpler 


§ix.] 


POLAR   COORDINATES  IN  SPACE. 


159 


forms  when  applied  to  solids  of  revolution.  Let  OZ,  Fig.  15, 
be  the  axis  of  revolution,  then  p  and  </;  are  polar  coordinates  of 
the  revolving  curve,  OR  being  the  initial  line.  Now  d  is  in 
this  case  independent  of  p  and  ^,  and  its  limits  are  o  and  27i. 
The  integration  with  reference  to  B  may  therefore  be  performed 
at  once.     Thus  from  (3)  we  obtain 


V  = 


27r 


ff  cos  4>  d^  ; 


(6) 


and  in  each  of  the  formulas  the  factor  2n  may  take  the  place 
of  the  integration  with  reference  to  d. 

140.  As  an  example  of  the  use  of  equation  (6),  let  us  find 
the  volume  generated  by  a  circle  revolving  about  one  of  its 
tangents.  The  initial  line,  being  perpendicular  to  the  axis  of 
revolution,  is  a  diameter;  hence  if  a  is  the  radius  of  the  circle 
its  equation  is 

p  ^=  2a  cos  (f), 

7t  7t 

and  the  limits  for  (j)  are and  — .     Substituting  in  (6) 


V  = 


iGna^ 


co?,*(j)  d^  —  2n^a^. 


141.  The  following  example  of  the  use  of 
equation  (4),  Art.  138,  is  added  to  illustrate 
the  necessity  of  drawing  a  figure  in  each 
case  to  determine  the  limits  to  be  employed. 

Let  it  be  required  to  find  the  volume 
generated  by  the  revolution  of  the  cardioid 
about  its  axis,  the  equation  of  the  curve 
being 

p  —  a{i  +  sin^),     .     . 


Fig.  16. 


(1) 


l6o  GEOMETRICAL  APPLICATIONS.  [Art.  I4I. 

when  the  initial  line  is  perpendicular  to  the  axis  of  the  curve, 
as  in   Fig.  16.      The   figure  shows  that  the  upper  limit  for  ^ 

is  -  n,  while  the  lower  limit  is  the  value  of  ^  given  by  equa- 
tion (1) ;  therefore 

sm02=i,  and  sm  pi  = 1. 

The  limits  for  p  are  evidently  o  and  2a.     Substituting  in  equa- 
tion (4)  Art.  138, 


=  27rr— -^'1"  =  ^-^— 
L  3        4«-io  3 


Examples  IX. 

I.  Find  the  volume  of  the  spheroid  produced  by  the  revolution  of 
the  ellipse, 

a-       0 
about  the  axis  of  x.  ~ . 


2.  Find  the  volume  of  a  right  cone  whose  altitude  is  a,  and  the 
radius  of  whose  base  is  b.  rtaP 


3.  Find  the  volume  of  the  solid  produced  by  the  revolution  about 
the  axis  of  .v  of  the  area  between  this  axis,  the  cissoid 

j'«  {2a  -  x)  =  .v', 
and  the  ordinate  of  the  point  (a,  a).  8aV  (log  2  —  |). 


IX.]  EXAMPLES.  l6l 

4.  Find  the  volume  generated  by  the  revolution  of  the  witch, 
y^x  —  2ay^  +  /^a^x  =  o, 


2_3 


about  its  asymptote. 

See  Art.  104.  47r*^ 

5.  The  equilateral  hyperbola 

X   —  y  =i  at 

revolves  about  the  axis  of  x  :  show  that  the  volume  cut  off  by  a  plane 
cutting  the  axis  of  .r  perpendicularly  at  a  distance  a  from  the  vertex 
is  equal  to  a  sphere  whose  radius  is  a. 

6.  An  anchor  ring  is  formed  by  the  revolution  of  a  circle  whose 
radius  is  l>  about  a  straight  line  in  its  plane  at  a  distance  a  froni  its 
centre  :  find  its  volume.  27t^ad\ 

7.  Express  the  volume  of  a  segment  of  a  sphere  in  terms  of  the 
altitude  /i  and  the  radii  Ui  and  Uc,  of  the  bases. 

~  {h'  +  za^  +  3a./). 

8.  Find  the  volume  generated  by  the  revolution  of  the  cycloid, 

A"  =  fl!  {ip  —  sin  ^'),  y  :=z  a  {i  —  cos  if:), 

about  its  base.  57rV. 

9.  The  area  included  between  the  cycloid  and  tangents  at  the 
cusp  and  at  the  vertex  revolves  about  the  latter  ;  find  the  volume  gen- 
erated. 

n  a  . 

10.  Find  the  volume  generated  by  the  revolution  of  the  part  of  the 

curve 

y^E-, 

which  is  on  the  left  of  the  origin,  about  the  axis  of  x. 

n 

2  * 


1 62  GEOMETRICAL   APPLICATIONS.  [Ex.  IX 

II.  The  axes  of  two  equal  right  circular  cylinders,  whose  common 
radius  is  a,  intersect  at  the  angle  a  ;  find  the  volume  common  to  the 
cylinders. 

T/te  section  Parallel  to  the  axes  is  a  rhombus.  i6<z' 

3  sin  a 

\z.  Find  the  volume  generated  by  the  revolution  of  one  branch  of 
the  sinusoid, 

a 

about  the  axis  of  x.  n^b 

2a  ' 

13.  Find  the  volume  enclosed  by  the  surface  generated  by  the  revo* 
lution  of  an  arc  of  a  parabola  about  a  chord,  whose  length  is  2r,  per- 
pendicular to  the  axis,  and  at  a  distance  b  from  the  vertex. 

14.  Find  the  volume  generated  by  the  revolution  of  the  tractrix, 
whose  differential  equation  is 

dv  V 


dx       -^  V{d'-f}' 
about  the  axis  of  -v. 

Express  ny'  dx  in  terms  of  y.  2  na^ 


15.  Find  the  volume  cut  from  a  right  circular  cylinder  whose  radius 
is  a,  by  a  plane  passing  through  the  centre  of  the  base,  and  making 
the  angle  a  with  the  plane  of  the  base. 

2c^  tan  a 

3 

16.  Fmd  the  volume  generated  by  the  curve 

.y:/  =  4^"  {2a  —  x) 
revolving  about  its  asymptote.  4?r*a'. 


§  IX.]  EXAMPLES.  163 

17.  Express  the  volume  of  a  frustum  of  a  cone  in  terms  of  its 
height  //,  and  the  radii  a^  and  ^2  of  its  bases. 

• —  («r+  a-^a,  +  a:). 
3 

18.  Find  the  volume  generated  by  the  revolution  of  the  cardioid, 

r  =  a  (i  —  C0S&), 
about  the  initial  line. 

Express  y  and  dx  in  terms  of  B.  8  na^ 

3 

19.  Find  the  volume  of  a  barrel  whose  height  is  2//,  and  diameter 
2b,  the  longitudinal  section  through  the  centre  being  a  segment  of  an 
ellipse  whose  foci  are  in  the  ends  of  the  barrel. 

2/^'  +  3<5' 


271  bVi 


3  {i^'  +  /n 


20.  Find  the  volume  generated  by  the  superior  and  by  the  inferioi 
branch  of  the  conchoid  each  revolving  about  the  directrix ;  the 
equation,  when  the  axis  of  j'  is  the  directrix,  being 


xy  ={a  +  xf  {P  -  x-"). 


....   ,   4^^' 


21.  On  two  opposite  lateral  faces  of  a  rectangular  parallelepiped 
whose  base  is  ab,  oblique  lines  are  drawn,  cutting  off  the  distances 
Ci,  Ci,  Cz,  Ci  on  the  lateral  edges.  A  straight  line  intersecting  each  of 
these  lines  moves  across  the  parallelopiped,  remaining  always  parallel 
to  the  other  lateral  faces  :  find  the  volume  cut  off. 

ab  (ci  +  Ci  +  ^3  +  ^4) 


22.  Find  the  volume  enclosed  by  the  surface  generated  by  an  arc 
of  a  circle  whose  radius  is  a,  about  a  chord  whose  length  is  2c. 

^^2 1  —  27ia  /^{a   —  b  )sm   '  — . 

2  <z 


164  GEOMETRICAL  APPLICATIONS.  [Ex.  IX. 

21.  The  area  included  between  a  quadrant  of  the  ellipse 

X  —  a  cos  ^,  y  =  b  sin  ^, 

and  the  tangents  at  its  extremities  revolves  about  the  tangent  at  the 
extremity  of  the  minor  axis  ;  find  the  volume  generated. 

nab''  (10  —  2,71) 


24.  An  ellipse  revolves  about  the  tangent  at  the  extremity  of  its 
major  axis  ;  express  the  entire  volume  in  the  fomi  of  an  integral, 
whose  limits  are  o  and  2  7r,  and  find  its  value.  zTi'a'b. 

25.  Show  that  the  volume  between  the  surface, 

s«  =  aKx"  +  by, 

and  any  plane  parallel  to  the  plane  of  -vy  is  equal  to  the  circumscrib- 
ing cylinder  divided  by  «  +  i. 

26.  A  straight  line  of  fixed  length  2C  moves  with  its  extremities  in 
two  fixed  perpendicular  straight  lines  not  in  the  same  plane,  and  at  a 
distance  zb.  Prove  that  every  point  in  the  moving  line  describes  an 
ellipse  in  a  plane  parallel  to  both  the  fixed  lines,  and  find  the  volume 
enclosed  by  the  generated  surface.  47r  (<:'  —  Ir)  b 


27.  Find  the  volume  enclosed  by  the  surface  whose  equation  is 

a        b       c  5 

28.  A  moving  straight  line,  which  is  always  perpendicular  to  a  fixed 
straight  line  through  which  it  passes,  passes  also  through  the  circum- 
ference of  a  circle  whose  radius  is  a,  in  a  plane  parallel  to  the  fixed 
straight  line  and  at  a  distance  b  from  it ;  find  the  volume  enclosed 
by  the  surface  generated  and  the  circle.  na^b 


Ttabc 

2 


§  IX.]  EXAMPLES.  165 

29.  Find  the  volume  enclosed  by  the  surface 

y'      ^'  _  -'^' 

Tii  "r   ~i 

oca 

and  the  plane  x  =  a. 

30.  Find  the  volume  enclosed  by  the  surface 

a.  a  2  y 

x^  +  jl^^  4-  s*  =  a^ , 

Find  A z  as  in  Art.  107,  a?id  then  evaluate  V  by  a  similar  jjiethod. 

47ra' 

31.  Find  the  volume  between  the  coordinate  planes  and  the  surface 

32.  Find  the  volume  cut  from  the  paraboloid  of  revolution 

y""  +  s''  =  4ax 
by  the  right  circular  cylinder 

x'  +  y-  =  2ax, 

whose  axis  intersects  the  axis  of  the  paraboloid  perpendicularly  at  the 

focus,  and  whose  surface  passes  through  the  vertex.  ,       i6a^ 

^  2na'  +  . 

3 
^2,.  The  paraboloid  of  revolution 

x-'  +/  =  cz 

is  pierced  by  the  right  circular  cylinder 

.v^  +  y""  =  ax, 


1 66  GEOMETA'ICAL  APPLICATIONS.  [Ex.  IX. 

whose  diameter  is  a,  and  whose  surface  contains  the  axis  of  the  parab- 
oloid ;  find  the  volume  between  the  plane  of  .r>'  and  the  surfaces  of 
the  paraboloid  and  of  the  cylinder.  Z'^a* 

34.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  a 
right  circular  cylinder  whose  radius  is  i,  and  whose  axis  passes  through 
the  centre  of  the  sphere.  47r 

3 


{a'-n'']. 


35.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve 

r  —  a  cos  30.  2a^7t        Sa' 

3  9 

36.  Find  the  volume  cut  from  a  sphere  whose  radius  is  a  by  the 
cylinder  whose  base  is  the  curve 

r  =  a  cos  0  +  0   sm  0, 

7   ^  47r^^       16  ,  J  ^3 

supposmg  b  <a. («  —  ")   • 

3  9 

37.  A  right  cone,  the  radius  of  whose  base  is  a  and  whose  alti- 
tude is  b^  is  pierced  by  a  cylinder  whose  base  is  a  circle  having  for 
diameter  a  radius  of  the  base  of  the  cone  ;  find  the  volume  common 

to  the  cone  and  the  cylinder.  bc^ ,  ,, 

-(97r-i6). 

38.  The  axis  of  a  right  cone  whose  semi- vertical  angle  is  ex  coin- 
cides with  a  diameter  of  the  sphere  whose  radius  is  a,  the  vertex  being 
on  the  surface  of  the  sphere  ;  find  the  volume  of  the  portion  of  the 
sphere  which  is  outside  of  the  cone.  4 tt^^  cos*  a 

3 

39.  Find  the  volume  produced  by  the  revolution  of  the  lemniscata 

r'  =  a^  cos  26, 

about  a  perpendicular  to  the  initial  line.  7t*a*  |/2 

8" 


§  IX.]  EXAMPLES.  167 

40.  Find  the  volumes  generated  by  the  revolution  of  the  large  loop 
and  by  one  of  the  small  loops  of  the  curve 

r  ^  a  cos  G  cos  29 
about  a  perpendicular  to  the  initial  line. 


—,2^3  ,— _3  _2^3  3 

7t  a        7ta         ,  7t  a        na 

— -, — I ,  and . 

16  5  32         10 


41.  From  the  element 

r  dr  dfj  dz 


derive  the  formulas  for  determining  the  volume  of  a  solid  of  revolution 
whose  axis  is  the  axis  of  z. 


V=z  2  7t  \    r  dr  dz , 

V—n\{r'^—rl)dz,  and  V  =  271    {z.,—z,)rdr. 

Interpret  the  elements  in  these  integrals. 

42.  Find  the  volume  generated  by  the  revolution  of  the  curve 

{x"  +  /y  =  a'x'  +  dy, 

in  which  a  >  b,  about  the  axis  oi  y. 

Transform  to  polar  coordinates,  and  use  the  method  of  Art.  139. 

6  ^  2'^{a'-b'y         a' 

43.  Find  the  volume  generated  by  the  curve  given  in  the  preceding 
example,  when  revolving  about  the  axis  of  x. 

Tta  {2a'  +  3b')  Tib"  a  +  \/{a''  -  i') 


1 68  GEOMETRICAL  APPLICATIONS.  [Ex.  IX. 

44.  Find  the  volume  common  to  the  sphere  whose  radius  is  p  =  a, 
and  to  the  solid  formed  by  the  revolution  of  the  cardioid, 

r  =  a  (i  4-  cos  O), 
about  the  initial  line. 

See  Art.  141.  —r~- 

45.  Find  the  whole  volume  enclosed  by  the  surface 

Transform  to  the  coordinates  p,  (/>,  0,  and  show  that  the  solid  consists 

a' 
0/ four  equal  detcuhed  parts.  —. 


X. 

Rectificatio7i  of  Plajic  Curves. 

142.  A  curve  is  said  to  be  rectified  when  its  length  is  deter- 
mined, the  unit  of  measure  to  which  it  is  referred  being  a 
right  line. 

It  is  shown  in  Diff.  Calc,  Art.  314  [Abridged  Ed.,  Art.  164], 
that,  if  s  denotes  the  length  of  the  arc  of  a  curve  given  in 
rectangular  coordinates,  we  shall  have 

ds  =  V{d.x^  +  df). 

If  the  abscissas  of  the  extremities  of  the  arc  are  known,  s  is 
found  by  substituting  for  dy  in  this  expression  its  value  in 
terms  of  x  and  dx,  and  integrating  the  result  between  the 
given  values  of  x  as  limits.  Thus,  to  express  the  arc  measured 
from  the  vertex  of  the  semi-cubical  parabola 

aji^=  x^ 


§  X.]  RECTIFICATION  OF  PLANE   CURVES.  l6g 

in  terms  of  the  abscissa  of  its  other  extremity,  we  derive,  from 
the  equation  of  the  curve, 


dy  — 


3  Vxdx 

2Va    ' 


whence 

2  Va 

Integrating, 

s  = 

\/(gx  +  4a)  dx 

2    |/rt    Jo 

I       ,  .a       %a 

= —  {gx  +  4^)' . 

27  Va  27 


143.  When  x  and  y  are  given  in  terms  of  a  third  variable, 
ds  is  generally  expressed  in  terms  of  this  variable.  For  exam- 
ple, from  the  equations  of  the  four-cusped  hypocycloid, 

X  —  a  cos^  ^',  y  —  a  sin^  ip,     .     .     .     (i) 

we  derive 
dx  =  —  3^  cos^  tp  sin  ip  dip,  and  dy  =  2^  sin^  ip  cos  ip  dip; 

whence  ds  =  T^a  sin  ip  cos  tp  dtp (2) 

The  length  of  the  arc  between  the  point  {a,  6),  corresponding 
to  ip  =  o,  and  (o,  a)  corresponding  to  ip  =  | tt,  is  therefore 


3^sins/ 
2 


3f 
2 


I/O  GEOMETRICAL  APPLICATIONS.  [Art.  1 44. 


Change  of  tJic  Sign  of  ds. 

(44.  Wc  have  hitherto  assumed  ds  to  be  positive,  but  it  is 
to  be  remarked  that  an  expression  substituted  for  ds^  as  in  the 
illustration  given  in  the  preceding  article,  may  change  sign. 
Thus,  in  equation  (2),  ds,  which  is  so  written  as  to  be  positive 
while  //•  passes  from  o  to  i/T,  becomes  negative  while  ^-  passes 
from  \7r  to  n.  Thus  the  integral  gives  a  negative  result  for 
the  arc  between  the  points  (o,  a)  and  {—a,  o),  corresponding  to 
\n  and  n.  This  change  of  sign  in  ds  indicates  a  cusp  or  sta- 
tionary point  of  the  curve ;  and  the  existence  of  such  points 
must  be  considered  before  we  can  properly  interpret  the  result- 
ing values  of  s.     For  instance,  if  in  this  example  we  integrate 

between  the  limits  o  and  —  ,  we  get  the  results  =  -- ,  which  is 

4  4 

the  algebraic   sum,    but    the    numerical  difference   of   the    arcs 
between  the  points  corresponding  to  the  limits. 


Polar  Coordinates. 

145.   It   is  proved   in   Diff.   Calc,  Art.  317  [Abridged    Ed., 
Art.  167],  that  when  the  curve  is  given  in  polar  coordinates 

ds  =  V(dr^  +  r^  d6^). 

This   is   usually  expressed   in   terms  of  B.     For  example,   the 
equation  of  the  cardioid  is 

r  —  a  {i  —  cos  B)  z=  2a  sin^ 4 0 ; 
whence  dr  =  2a  sin  -jB  cos  ^0  d6, 

and  by  substitution 

ds  =  2a  sin  |  /V  dB. 


§  X.]  RECTIFICATION  OF  CURVES.  I7I 

The  limits  for  the  whole  perimeter  of  the  curve  are  o  and  27r, 
and  ds  remains  positive  for  the  whole  interval.     Therefore 


s  =■  2a\    sin  —dd^=  —4a cos 

Jo  2 


=  8^. 


Rectification  of  Curves  of  Dotcble  C^crvattcre. 

146.  Let  ff  denote  the  length  of  the  arc  of  a  curve  of  double 
curvature ;  that  is,  one  which  does  not  lie  in  a  plane,  and  sup- 
pose the  curve  to  be  referred  to  rectangular  coordinates  ;f,  j^ 
and  ^'.  If  at  any  point  of  the  curve  the  differentials  of  the 
coordinates  be  drawn  in  the  directions  of  their  respective  axes, 
a  rectangular  parallelopiped  will  be  formed,  whose  sides  are 
dxydy  and  ds^  and  whose  diagonal  is  da.     Hence 

da  =  V{dx^  +  df  +  d:?). 

The  curve  is  determined  by  means  of  two  equations  connect- 
ing .r,  y  and  z,  one  of  which  usually  expresses  the  value  of  y  in 
terms  of  x,  and  the  other  that  of  z  in  terms  of  x.  We  can 
then  express  d(S  in  terms  of  x  and  dx. 

If  the  given  equations  contain  all  the  variables,  equations 
of  the  required  form  may  be  obtained  by  elimination. 

147.  An  equation  containing  the  two  variables  x  and  y 
only  is  evidently  the  equation  of  tJie  projection  upon  the  plane 
of  xy  of  a  curve  traced  upon  the  surface  determined  by  the 
other  equation.  Let  j-  denote  the  length  of  this  projection; 
then,  since  ds^  =  dx^  +  df', 

d^  =  V{ds^  +  d^), 

in  which  ds  may,  if  convenient,  be  expressed  in  polar  coordin- 
ates ;  thus, 

d(j=  V{dr^  "^  r^dS'^  ^  dz% 


172  GEOMETRICAL  APPLICATIONS.  [Art.   1 48. 

!48.  As  an  illustration,   let  us  use  this  formula  to   deter- 
mine the  length  of  the  loxodromic  curve  from  the  equation  of 

the  sphere, 

x"  ^  f  ^  :?  =  a', (I) 

upon  which  it  is  traced,  and  its  projection  upon  the  plane  of 
the  equator,  of  which  the  equation  is 

2a  -  ^{x^  +  /)  ii."  '''>"-'!^  +  f-«  '^"  ij  , 

or  in  polar  coordinates 

2a  =  r  («""  +  f-'-") (2) 

Equation  (i)  is  equivalent  to 

r=  +  ^  =  ^2 ; 

and,  denoting   the  latitude  of  the  projected  point  by  ^,  this 

gives 

^  =  rt'  sin  ^,  r  =  a  cos  ^.     .     .     .     (3) 

In  order  to  express  dQ  in  terms  of  ^,  we  substitute  the  value 
of  r  in  (2)  ;  whence 

f  «e  J.  f  -  «e  —  2  sec  ^, (4) 

and  by  differentiation 

£""  -  f-«^  =  -  sec  ^  tan  c5  -J? (5) 

;/  dO 

Squaring  and  subtracting  equation  (5)  from  equation  (4), 

4  sec^^r  2  iA^^~\ 

4-^^L.^-tan^^^-^J, 


which  reduces  to 


de''  =  ^^l±^ (6) 


§  X,]  LENGTH  OF   THE  LOXODROMTC  CURVE.  1/3 

From  equations  (3)  and  (6) 

dz^  =  cr  cos^  ^  d<^ ; 
whence  substituting  in  the  value  of  dG  (p.  171) 

d6=:av{i+^d^. 
Integrating, 

G  =  a—^ '\    d<p  —  a—^ '  {6  —  a), 

n         Ja  n  ' 

where  a  and  /3  denote   the    latitudes   of   the    extremities   of 
the  arc. 

Examples  X. 

1.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 
catenary 

X  X 

and  show  that  the  area  between  the  coordinate  axes  and  any  arc  is 
proportional  to  the  arc, 

,      f.  X 

.  =  !(/-.-). 

A  =  cs. 

2.  Find  the  length  of  an  arc  measured  from  the  vertex  of  the 
X)arabola 

^x  +  \/{x  +  a) 


V{ajc  -i-  x^)  +  a\o^ 


Va 


174  GEOMETRICAL  APPLICATJOxVS.  [Ex.  X, 

3.  Find  the  length  of  the  curve 

f-'  +  I 
f-^  —  I ' 

between  the  points  whose  abscissas  are  a  and  b. 

1      f  "^  —  I 

log \-  a  —  0. 

4.  Find  the  length,  measured  from  the  origin,  of  the  curve 

a   —  X 


y  —  a  log 


a  log X. 

°  a  —  X 


5.  Given  the  differential  equation  of  the  tractrix, 

^-        y 

dx  '^{a^-yy 

and,  assuming  (o,  a)  to  be  a  point  of  the  curve,  find  the  value  of  s  as 
measured  from  this  point,  and  also  the  value  of  x  in  terms  oiy  ;  that 
is,  find  the  rectangular  equation  of  the  curve. 

y 
s  —  a  log  — . 

x  =  a\o% ^— \'[a  —  y). 

y 

6.  Find  the  length  of  one  branch  of  the  cycloid 

X  z=z  a{tp  —  sin  ^'),  y  z=  a  {i  —  cos  r(^). 

8a. 

7.  When  the  cycloid  is  referred  to  its  vertex,  the  equations  being 

-v  =  a  (i  —  cos  ip),  y  z=  a{'p  +  sin  ip), 

prove  that  s  =  v/(8a-v). 


§  X.]  EXAMPLES.  175 

8.  Find  the  length  from  the  point  {a,  o)  of  the  curve 

X  =  2a  cos  ?/'  —  a  cos  2^, 

y  ■=^  2a  sin  tp  —  a  sin  2?/?. 

4a  ('/'  —  sin  ^). 

9.  Show  that  the  curve, 

a:  =  3  «  cos  ip  —  2a  cos'  ^,  _y  =  2^z  sin'  ^, 

has  cusps  at  the  points  given  by  ?/>  =  o  and  tp  =  rr ;  and  find  the 
whole  length  of  the  curve.  12a. 

10.  Find  the  length  of  a  quadrant  of  the  curve 

r      r-     c    A  *                                                             d'  +  ad  +  F 
See  Fis".  6,  Art.  107. —  . 

11.  Show  that  the  curve 
X  ^=^  2a  cos'''  ^  (3  —  2  cos"  6),  y  =  ^a  sin  0  cos'  (9 


has  three  cusps,  and  that  the  length  of  each  branch  is  • —  . 

3 

12.  Find  the  length  of  the  arc  between  the  points  at  which   the 
curve 

X  =  a  cos''  0  cos  26,  _^  =  «  sin  ^  ^  sin  2  ^ 

2  —  V2 
cuts  the  axes.  a. 


176  GEOMETRICAL   APPLICATIONS.  [Ex.  X. 

13.  Show  that  the  curve 

-v  =  a  cos  ^"  (i  +  sin''  </•), 
y  ^=  a  sin  '/'  cos'  '/• 

is  symmetrical  to  the  axes,  and  find  the  length  of  the  arcs  between 
the  cusps.  /    .  .  I  \ 

\  V3/ 

rt!  (    ^2   +  cos-'  

14,  Find  the  length  of  one  branch  of  the  epicycloid 

/         ,v         /       >        a  -\r  b  , 
x=  (a  -V  h)  cos  ip  —  0  cos  — - —  ?/', 


y  ^=.  {a  -\-  b)  sm  f  —  b  sm  — 7—  ip. 


U  {a  +  b) 


15.  Show  that  the  curve 

.V  =  9<2  sin  y,'  —  4a  sin'  '/•, 
y  ^  —  T^a  cos  '/•  +  4a  cos'  '/• 

is  symmetrical  to  the  axes,  and  has  double  points  and  cusps  :  find  the 
lengths  of  the  arcs,  (a)  between  the  double  points,  (/S)  between  a 
double  point  and  a  cusp,  and  {y^  the  arc  connecting  two  cusps,  and  not 
passing  through  the  double  points. 

(«,  T^ 

16.  Find  the  whole  length  of  the  curve 

.V  =  3<z  sin  '/.'  —  a  sin'  ^, 

y=  a  cos'  tp.  3  7Ta. 


§  X.]  EXAMPLES.  177 

17.  Find  the  length,  measured  from  the  pole,  of   any  arc   of  the 
equiangular  spiral 

r  =  as"^, 
in  which  n  =  cot  a.  r  sec  a. 

18.  Prove  by  integration  that  the  arc  subtending  the  angle  d  at  the 
circumference  in  a  circle  whose  radius  is  a,  is  2«(5. 

19.  Find  the  length,  measured  from  the  origin,  of  the  curve  defined 
by  the  equations 


X  X 


60" 


x^ 
oa' 


20.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  surfaces 

y  =  4u  sin  .r,  z  =^  2^1^  {2X  +  sin  2x). 

(4«^  +  i)x  +  2;^"  sin  2X. 

21.  Find  the  length,  measured  from  the  origin,  of  the  intersection  of 
the  cylindrical  surfaces 


2Jt:2 


22.  If  upon  the  hyperbolic  cylinder 


+  2  V{ax)+x. 


i  - ;;  = ., 

a  curve  whose  projection  upon  the  plane  of  xy  is  the  catenary 

X  X 

_j;=-(f     +6        ) 

be  traced,  prove  that  any  arc  of  the  curve  bears  to  the  corresponding 
arc  of  its  projection  the  constant  ratio  '^{b'^  +  c^)  :  c. 


178 


GEOMETRICAL   APPLICATIONS. 


[Art.  149 


XI. 


Siw faces  of  Solids  of  Revolution. 

149.  The  surface  of  a  solid  of  revolution  may  be  generated 
by  the  circumference  of  the  circular  section  made  by  a  plane 

perpendicular  to  the  axis  of  revolu- 
tion. Thus  in  Fig.  17,  the  surface 
produced  by  the  revolution  of  the 
curve  AB  about  the  axis  of  x  is  re- 
garded as  generated  by  the  circum- 
ference PQ.  The  radius  of  this  cir- 
cumference is  y,  and  its  plane  has  a 
motion  whose  differential  is  dx\  but 
every  point  in  the  circumference  itself 
has  a  motion  whose  differential  is  ds^s 
denoting  an  arc  of  the  curve  AB. 
Hence,  denoting  the  required  surface  by  S,  we  have 

dS=  27Ty  ds  =  271  y  V{d.v^  +  dj^). 

The  value  of  dS  must  of  course  be  expressed  in  terms  of  a  single 
variable  before  integration. 

150.  For  example,  let  us  determine  the  area  of  the  zone  of 
spherical  surface  included  between  any  two  parallel  planes. 
The  radius  of  the  sphere  being  a,  the  equation  of  the  revolv- 
ing curve  is 


+  /  r=   ^2  . 


whence 


dy  = 


ds  = 


Via'-x"), 
X  dx 


a  dx 


^y 


and 


^{tV"  —  x^) 
dS  =  27ra  dx ; 


XI.]  SURFACES    OF   SOLIDS    OF  REVOLUTION.  1 79 


therefore 


S  ^  27ia\  dx  =  2  TTrt  {xi  —  Xi)  . 


Since  Xo  —  Xi  is  the  distance  between  the  parallel  planes, 
the  area  of  a  zone  is  the  product  of  its  altitude  by  27ra,  the 
circumference  of  a  great  circle,  and  the  area  of  the  whole  sur- 
face of  the  sphere  is  47Tir. 

151.  When  the  curve  is  given  in  polar  coordinates,  it  is  con- 
venient to  transform  the  expression  for  5  to  polar  coordinates. 
Thus,  if  the  curve  revolves  about  the  initial  line, 

S  =  27r\j/  ds=  27t  r  sin  d  \ \dr^  +  r^  dd^). 


For  example,  if  the  curve  is  the  cardioid 
we  find,  as  in  Art.  145, 


r  —2o-  sin^—  0  , 
2 


ds  =  2a  sin  —  6  dd. 

2 


Hence 


S  =  i67ra^\  sin^-6'cos- 
Jo        2  2 


^  cos  -  ^  dd 


%27Ta^    .   -  I 
^ sm^- 


327rrt' 


Areas  of  Surfaces  in  General. 

152.  Let  a  surface  be  referred  to  rectangular  coordinates  x, 
y  and  z ;  the  projection  of  a  given  portion  of  the  surface  upon 
the  plane  of  xy  is  a  plane  area  determined  by  a  given  relation 
between  x  and  y.  We  may  take  as  the  elements  of  the  surface 
the   portions   which   are   projected    upon    the    corresponding 


i8o 


GEO  ME  TRIG  A  L  A  P  PLICA  TIOXS. 


[Art.  152. 


elements  of  area  in  the  plane  of  xy.  If  at  a  point  within  the 
clement  of  surface,  which  is  projected  upon  a  given  element 
AxAjy,  a  tangent  plane  be  passed,  and  if  y  denote  the  inclina- 
tion of  this  plane  to  the  plane  of  .ij,  the  area  of  the  correspond- 
ing element  in  the  tangent  plane  is 

sec  y  A  X  A  J'. 

The  surface  is  evidently  the  limit  of  the  sum  of  the  elements 
in  the  tangent  planes  when  A.r  and  l\y  are  indefinitely  dimin- 
ished. Now  sec  y  \%  z.  function  of  the  coordinates  of  the  point 
of  contact  of  the  tangent  plane  ;  and  since  these  coordinates 
are  values  of  x  and  y  which  lie  respectively  between  x  and 
X  -I-  A.f  and  between  J  and  j'  +  aj',  the  theorem  proved  in  Art. 
99  shows  that  this  limit  is 


vS  =      sec  y  dx  dy. 

153.  The  value  of  sec  ;'   may  be   derived  by   the  following 

method.     Through  the  point  P  of 
the  surface  let   planes   be   passed 
parallel  to  the    coordinate    planes, 
and  let  PD,  and  PE,  Fig.  17,  be  the 
intersections  of  the  tangent   plane 
with    the   planes   parallel    to    the 
planes  of  xz  and  j^.     Then  PD  and 
PE  are  tangents  at  P  to  the  sec- 
tions of  the  surface  made  by  these 
planes.     The    equations   of    these 
sections  are  found  by  regarding  y 
and  X  in  turn  as  constants  in  the  equation  of  the  surface  ;  there- 
fore denoting  the  inclinations  of  these  tangent  lines  to  the  plane 
of  xy  by  ^  and  ?/•,  we  have 

.       dz  ,  ,        dz 

tan  (p—  ---        ■         and  tan  v  =  -^ , 

dx  dy 


Fig.  18. 


§  XL]  AREAS  OF  SURFACES  IN  GENERAL.  l8l 

in  which  — -  and^  are  partial  derivatives  derived  from  the  equa- 
tor        ay  '■ 

tion  of  the  surface. 

If  the  planes  be  intersected  by  a  spherical  surface  whose 
centre  is  P,  ADE  is  a  spherical  triangle  right  angled  at  A, 
whose  sides  are  the  complements  of  ^  and  ?/?.  Moreover,  if  a 
plane  perpendicular  to  the  tangent  plane  PED  be  passed 
through  AP,  the  angle  FPG  will  be  ;/,  and  the  perpendicular 
from  the  right  angle  to  the  base  of  the  triangle  the  comple- 
ment of  y. 

Denoting  the  angle  EAF  by  d,  the  formulas  for  solving 
spherical  right  triangles  give 

r.      tan  ?/'  J  .     ^      tan  ^ 

cos  u  = ,  and  sm  a  = . 

tan  y  tan  y 


Squaring  and  adding, 


I  = 


tan^  tfj  +  tan^  ^ 


tan'*  y 
or  tan^  y  =  tan^  ip  +  tan^  ^ ; 


whence  secV  =  i  +  [-j;J    +  [jj,)  • 


Substituting  in  the  formula  derived  in  Art.  (152),  we  have 

154.  It  is  sometimes  more  convenient  to  employ  the  polar 


1 82  GEOMETRICAL  APPLICATIONS.  [Art.  1 54, 

cicmcnt  of  the  projected  area.     Thus  the  formula  becomes 

5  =      sec  yrdrdS, 

where  sec  y  has  the  same  meaning  as  before. 

For  example,  let  it  be  required  to  find  the  area  of  the  sur- 
face of  a  hemisphere  intercepted  by  a  right  cylinder  having  a 
radius  of  the  hemisphere  for  one  of  its  diameters.  From  the 
equation  of  the  sphere, 


we  derive 


whence 


x"  +y^  -^  :^=d\ (i) 


dz_       X  dz  _      jj' 

dx  2^  dy  -s   ' 


sec}/=  j^ 


/m/(IT+(|)V 


.  Irdrdd 
therefore 


-"IP 


the  integration  extending  over  the  area  of  the  circle 

r  =  a  cos  6 (2) 

Since  equation  (i)  is  equivalent  to 

^  +  r^  =  ^, 


I  XL]  AREAS  OF  SURFACES  IN  GENERAL.  1 83 

From  (2)  the  limits  for  r  are  ri  =  o,  and  r2  =  a  cos  6, 
hence 


5  =  rt2f(^i  -sme)dd, 


in  which  a  sin  B  is  put  for  the  positive  quantity  V{a^  —  r.^).  The 
limits  for  6  are  —  Itt  and  Itt,  but  since  sin  6  is  in  this  case  to 
be  regarded  as  invariable  in  sign,  we  must  write 


S-=  2a'\\i-  sin  6)  dd  =  na^ -  2a\ 


If  another  cylinder  be  constructed,  having  the  opposite  radius 
of  the  hemisphere  for  diameter,  the  surface  removed  is 
27ra^  —  4a^,  and  the  surface  which  remains  is  40^,  a  quantity 
commensurable  with  the  square  of  the  radius.  This  problem 
was  proposed  in  1692,  in  the  form  of  an  enigma,  by  Viviani,'a 
Florentine  mathematician. 


Examples  XI. 

I.  Find  the  surface  of  the  paraboloid  whose  altitude  is  a,  and  the 
radius  of  whose  base  is  l>. 


2.  Prove  that  the  surface  generated  by  the  arc  of  the  catenary  given 
in  Ex.  X.,  I,  revolving  about  the  axis  of  .v,  is  equal  to 

7r{cx  +  J}'). 

3.  Find  the  whole  surface  of  the  oblate  spheroid  produced  by  the 


1 84  GEOMETRICAL  APPLICATIOXS.  [Ex.  XI. 

revolution  of  an  ellipse  about  its  minor  axis,  a  denoting  the  major, 
b  the  mmor  semi-axis,  and  e  the  excentncity, . 

.   .        b\      I  +^ 

2na    +  TT  -  log . 

e         1  —  e 

4.  Find  the  whole  surface  of  the  prolate  spheroid  produced  by  the 
revolution  of  the  ellipse  about  its  major  axis,  using  the  same  notation 
as  in  Ex.  3. 

,  iSin-V 

2  7ti?    +  271  ab  . 


5.  Find  the  surface  generated  by  the  cycloid 

X  =  a  {ip  —  sin  '/),         y  =^  a  {i  —  cos tp) 

revolving  about  its  base.  —  tco'. 

3 

6.  Find  the  surface  generated  when  the  cycloid  revolves  about  the 
tangent  at  its  vertex. 

3 

7.  Find  the  surface  generated  when  the  cycloid  revolves  about  its 


\7Ta'{    Tt   —  - 


8.  Find  the  surface  generated  by  the  revolution  of  one  branch  of 
the  tractrix  (see  Ex.  X.,  5)  about  its  asymptote. 

2  /Ta^ 


§  XL]  EXAMPLES.  185 

9.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 
X  of  the  portion  of  the  curve 


y  =  f ", 

which  is  on  the  left  of  the  axis  of  j'. 


7r[Y2  +  log  (i  +    12)]. 


10.  Find  the  surface  generated  by  the  revolution  about  the  axis  of 
-v  of  the  arc  between  the  points  for  which  x  =  a  and  x  —  i>  in  the 
hyperbola 

xy  =  k\ 


7tk' 


'       P  +  \'{k'  +  //)        ^/{k'  +  a')  _  six  +  b') 


II.  Show  that  the  surface  of  a  cylinder  whose  generating  lines  are 
parallel  to  the  axis  of  z  is  represented  by  the  integral 

6"  =  \z  iis , 


where  s  denotes  the  arc  of  the  base  in  the  plane  of  xy.  Hence, 
deduce  the  surface  cut  from  a  right  circular  cylinder  whose  radius  is 
a,  by  a  plane  passing  through  the  centre  and  making  the  angle  (v  with 
the  plane  of  the  base.  20^  tan  a. 

12.  Find  the  surface  of  that  portion  of  the  cylinder  in  the  problem 
solved  in  Art.  154,  which  is  within  the  hemisphere.  2«^ 

13.  Find  the  surface  of  a  circular  spindle,  a  being  the  radius  and 
2c  the  chord. 


^7ta 


c  —   s/ia   —  ^')sin-'- 
a 


1 86  GEOMETRICAL  APPLICATIONS.  [Art.  1 5 5. 


XII. 

TJic  Area  generated  by  a  Straight  Line  moving  in  any 
Manner  i)i  a  Plane. 

155.  If  a  straight  line  of  indefinite  length  moves  in  any  man- 
ner whatever  in  a  plane,  there  is  at  each  instant  a  point  of  the 
line  about  which  it  may  be  regarded  as  rotating.  This  point  we 
shall  call  the  centre  of  rotation  for  the  instant.  The  rate  of 
motion  of  every  point  of  the  line  in  a  direction  perpendic- 
ular to  the  line  itself  is  at  the  instant  the  same  as  it  would 
be  if  the  line  were  rotating  at  the  same  angular  rate  about  this 
point  as  a  fixed  centre.*  Hence  it  follows  that  the  area 
generated  by  a  definite  portion  of  the  line  has  at  the  instant 
the  same  rate  as  if  the  line  were  rotating  about  a  fixed  instead 
of  a  variable  centre. 

(56.  Suppose  at  first  that  the  centre  of  rotation  is  on  the 
generating  line  produced,  pj  and  p^  denoting  the  distances  from 
the  centre  of  the  extremities  of  the  generating  line,  and  let  ^ 
denote  its  inclination  to  a  fixed  line.  By  substitution  in  the 
general  formula  derived  in  Art.  no,  we  have 

dA  =  ^-  {pi-  pi)  d(j>. 

*  Compare  Diff.  Calc,  Art.  332  [Abridged  Ed.,  Art.  176J,  where  the  moving 
linfe  is  the  normal  to  a  given  cur\e,  and  the  centre  of  rotation  is  the  centre  of  cur- 
vature of  the  given  curve.  If  the  line  is  moving  without  change  of  direction,  the 
centre  is  of  course  at  an  infinite  distance. 

Wiien  the  line  is  regarded  as  forming  a  part  of  a  rigidly  connected  system  in 
motion,  its  centre  of  rotation  is  the  foot  of  a  perpendicular  dropped  upon  it  from 
the  insUxutaueous  centre  of  the  motion  of  the  system.  Thus,  if  the  tangent  and 
normal  in  the  illustration  cited  are  rigidly  connected,  the  centre  of  curvature,  C,  is 
the  instantaneous  centre  of  the  motion  of  the  system,  and  the  point  of  contact,  P, 
is  the  centre  of  rotation  for  the  tangent  line. 


XII.]  AREAS  GENERATED  BY  MOVING  LINES.  1 8/ 


Applications. 

157.  The  area  between  a  curve  and  its  evolute  may  be 
generated  by  the  radius  of  curvature  p,  whose  incHnation  to 
the  axis  of  ;ir  is  ^  +  \7t,  in  which  ^  denotes  the  inclination 
of  the  tangent  line.  Since  the  centre  of  rotation  is  one 
extremity  of  the  generating  line/?,  the  differential  of  this  area 
is  found  by  substituting  in  the  general  expression  i\  =  o  and 
/32  =  o.     Hence  when  p  is  expressed  in  terms  of  ^, 


A 


=  -  \/rd(^ 


expresses  the  area  between  an  arc  of  a  given  curve,  its  evolute, 
and  the  radii  of  curvature  of  its  extremities,  the  limits  being 
the  values  of  (/>  at  the  ends  of  the  given  arc. 

158.  For  example,  in  the  case  of  the  cardioid 

r  =  a{i  —  cos  6), 

it  is  readily  shown,  from  the  results  obtained  in  Art.  145,  that 
the  angle  between  the  tangent  and  the  radius  vector  is  ^6^;  and 
therefore  ^  =  f  ^,  and 

ds       Aa   .    6 

d<!>       3        3 

To  obtain  the  whole  area  between  the  curve  and  its  evolute, 
the  limits  for  0  are  O  and  2/T  ;  hence  the  limits  for  ^  are  o 
and  37r.     Therefore 


2 


8^2 


f?  d(l) 

9 


(f)    , ,      ^7td 


sin^  ~  d(l)  = 
3 


159.  As    another   application    of    the    general    formula    of 
Art.  156,  let  one  end   of  a  line  of  fixed  length  a  be  moved 


1 88  GEOMETRICAL   APPLICATIONS.  [Art.   1 59. 

along  a  given  line  in  a  horizontal  plane,  while  a  weight  at- 
tached to  the  other  extremity  is  drawn  over  the  plane  by  the 
line,  and  is  therefore  always  moving  in  the  direction  of  the 
line  itself.  The  line  of  fixed  length  in  this  case  turns  about 
the  weight  as  a  moving  centre  of  rotation.  Hence  the  area 
generated  while  the  line  turns  through  a  given  angle  is  the 
same  as  that  of  the  corresponding  sector  of  a  circle  whose 
radius  is  a. 

The  curve  described  b)'  the  weight  is  called  a  tractrix,  and 
the  line  along  which  the  other  extremity  is  moved  is  tJie  direc- 
trix. When  the  axis  of  x  is  the  directrix,  and  the  weight 
starts  from  the  point  (o,  a),  the  common  tractrix  is  described ; 
hence  the  area  between  this  curve  and  the  axis  is  \Ttc?. 

160.  Again,  in  the  generation  of  the  cycloid,  Diff.  Calc, 
Art.  288  [Abridged  Ed.,  Art.  156],  the  variable  chord  RP  may 
be  regarded  as  generating  the  area.  The  point  R  has  a  motion 
in  the  direction  of  the  tangent  RX \  the  point  /'partakes  of 
this  motion,  which  is  the  motion  of  the  centre  C,  and  also  has 
an  equal  motion,  due  to  the  rotation  of  the  circle  in  the  direc- 
tion of  the  tangent  to  the  circle  at  P.  Since  the  tangents 
at  P  and  R  are  equally  inclined  to  PR,  the  motion  of  P  in  a 
direction  perpendicular  to  PR  is  double  the  component,  in  this 
direction,  of  the  motion  of  R.  Therefore  the  centre  of  rota- 
tion of  PR  is  beyond  v'?  at  a  distance  from  it  equal  to  PR. 
Hence,  denoting  PRO  by  <5, 

pj  —  PR  —  2a  sin  ^,  />2  =  2PR  =  4a  sin  ^. 

Substituting  in  the  formula  of  Art.  156,  we  have  for  the  area 
of  the  cycloid,  since  PRO  varies  from  o  to  n. 


A  = 


=  6a^     sin^  <p  d(p  =  ^Tta^. 


§  XII.] 


SIGN   OF    THE    GENERATED  AREA. 


189 


Fig.  19. 


Sign  of  the  Generated  Area. 

161.  Let  AB  be  the  generating  line,  and  6"  the  centre  of 

rotation.     The  expression, 

dA  ^^,{pi-p^)d<f>, (I) 

for  the  differential  of  the  area,  was  obtained  upon  the  supposi- 
tion that  A  and  B  were  on  the  same  side  of  C.     Then  suppos- 
ing P2  >  Px,  and  that  the  line  rotates  in  the  positive  direction^ 
as  in  figure  19,  the  differential  of  the  area  is 
positive;  and  we  notice  that  every  point  in  the 
area  generated   is  swept    over  by  the  line 
AB,    the  left  hand  side  as  we  face   in  the 
direction  A  B  preceding. 

162.  We  shall  now  show  that  in  every 
case,  the  formula  requires  that  an  area 
swept  over  with  the  left  side  preceding,  shall  be  considered 
as  positively  generated,  and  one  swept  over  in  the  opposite 
direction  as  negatively  generated. 

In  the  first  place,  if  C  is  between  A  and 
B  so  that  Pi  is  negative,  as  in  figure  20,  pi 
is  still  positive,  and  formula  (i)  still  gives 
the  difference  between  the  areas  generated 
by  ^;5  and  AC.  Hence  the  latter  area, 
which  is  now  generated  by  a  part  of  the 
line  AB,  must  be  regarded  as  generated 
negatively,  but  the  right  hand  side  as  we 
face  in  the  direction  AB  of  this  portion  of  the  line  is  nov/ 
preceding,  which  agrees  with  the  rule  given  in  Art.  161. 

Again,  if  C  is  beyond  B,  the  formula  gives  the  difference 
of  the  generated  areas;  but  since  pi  is  numerically  greater 
than  pI,  in  this  case,  dA  is  negative,  and  the  area  generated  by 
AB  is  the  difference  of  the  areas,  and  is  negative  by  the  rule. 


Fig.  20. 


190 


GEOMETRICAL  APPLICATIONS. 


[Art.  162. 


Finally,  if  the  direction  of  rotation  be  reversed,  d^  and 
therefore  dA  change  sign,  but  the  opposite  side  of  each  por- 
tion of  the  line  becomes  in  this  case  the  preceding  side. 

163.  We  may  now  put  the  expression  for  the  area  in  another 
form.     For 


dA=-  (p^ 

2  ^  ' 


Pi)  i^4>  =  (P2 


^,)a±Arf^: 


whatever  be  the  signs  of  p^  and  Pj,  the  first  factor  is  the  length 
of  AB,  which  we  shall  denote  by  /,  and  the  second  factor  is 
the  distance  of  the  middle  point  of  AB  from  the  centre  of 
rotation,  which  we  shall  denote  by  p^.     Hence,  putting 


Ih  -  Pi  =  I, 
we  have 


and 


A 


=    Upmd^. 


Pl±Jh-   n 
2  ~^'"' 


(2) 


Since  p,„d(p  is  the  differential  of  the  motion  of  the  middle  point 
in  a  direction  perpendicular  to  AB,  this  expression  shows  that 
the  differential  of  the  area  is  the  product  of  this  differential  by 
the  length  of  the  generating  line. 


Areas  generated  by  Lines  whose  Extremities   describe 
Closed  Circtiits. 


i^* 


164.  Let  us  now  suppose  the  generating  line  AB  to  move 
from  a  given  position,  and  to  return  to  the 
same  position,  each  of  the  extremities  A  and 
B  describing  a  closed  curve  in  the  positive 
direction,  as  indicated  by  the  arrows  in  figure 
21.  It  is  readily  seen  that  every  point  which 
is  in  the  area  described  by  B,  and  not  in  that 
described  by  A,  will  be  swept  over  at  least 
once  by  the  line  AB,  the  left  side  preceding, 

Fig.  21.  and  if  passed  over  more  than  once,  there  will  be 


§  XII.]        AREAS  GENERATED  BY  MOVING  LINES.  I9I 

an  excess  of  one  passage,  the  left  side  preceding.  Therefore 
the  area  within  the  curve  described  by  B,  and  not  within  that 
described  by  A,  will  be  generated  positively.  In  like  manner 
the  area  within  the  curve  described  by  A,  and  not  within  that 
described  by  i?,  will  be  generated  negatively.  Furthermore,  all 
points  within  both  or  neither  of  these  curves  are  passed  over, 
if  at  all,  an  equal  number  of  times  in  each  direction,  so  that  the 
area  common  to  the  two  curves  and  exterior  to  both  disap- 
pears from  the  expression  for  the  area  generated  by  AB. 

Hence  it  follows  that,  regarding  a  closed  area  zuJiose  perimeter 
is  described  in  the  positive  direction  as  positive,  the  area  generated 
by  a  line  returning  to  its  original  position  is  the  differe?tce  of  the 
areas  described  by  its  extremities.  This  theorem  is  evidently 
true  generally,  if  areas  described  in  the  opposite  direction  are 
regarded  as  negative. 


Amslers  Planimeter. 

165.  The  theorem  established  in  the  preceding  article  may 
be  used  to  demonstrate  the  correctness  of  the  method  by 
which  an  area  is  measured  by  means  of  the  Polar  Planimeter, 
invented  by  Professor  Amsler,  of  Schaffhausen. 

This  instrument  consists  of  two  bars,  OA  and  AB,  Fig.  22, 
jointed  together  at  A.  The  rod  OA  turns  on 
a  fixed  pivot  at  O,  while  a  tracer  at  B  is  carried 
in  the  positive  direction  completely  around 
the  perimeter  of  the  area  to  be  measured.  At 
some  point  C  of  the  bar  AB  a  small  wheel  is 
fixed,  having  its  axis  parallel  to  AB,  and  its 
circumference  resting  upon  the  paper.  When 
^is  moved,  this  wheel  has  a  sliding  and  a  roll- 
ing motion  ;  the  latter  motion  is  recorded  by 
an  attachment  by  means  of  which  the  number  Fig.  22. 

of  turns  and  parts  of  a  turn  of  the  wheel  are  registered. 


192  GEOMETRICAL  APPLICATIONS.  [Art.  1 66. 

166.  Let  J/ be  the  middle  point  of  AB,  and  let 
OAr^a,  AB  =  b,  MC  =  c. 

Since  b  is  constant,  the  area  described  by  yJ/>  is  by  equation  (2), 
Art.  163, 

Area^i5  =  ^|p,„^/^ (i) 

Denoting  the  linear  distance  registered  on  the  circumference 
of  the  wheel  by  s,  ds  is  the  differential  of  the  motion  of  the 
point  C,  in  a  direction  perpendicular  to  AB,  and  since  the  dis- 
tance of  this  point  from  the  centre  of  rotation  is  f>„i  +  c, 

ds  =  {p„,  +  c)  d^: 
substituting  in  (i)  the  value    of  p„,d(f>, 

-  bc[d(}> (2) 


Area  AB  —  b 


ds 


167.  Two  cases  arise  in  the  use  of  the  instrument.  When, 
as  represented  in  Fig.  22,  O  is  outside  the  area  to  be  meas- 
ured, the  point  A  describes  no  area,  and  by  the  theorem  of 
Art.   164,   equation   (2)   represents  simply   the  area   described 


by  B.     In  this  case  ^  returns  to  its  original   value,   hence 


d(!> 


vanishes,  and  denoting  the  area  to  be  measured  by^,  equation 
(2)  becomes 

A  =  bs (3) 

In  the  second  case,  when  O  is  within    the   curve  traced  by  B, 
the  point  A  describes  a  circle  whose   area  is  nd^,  and  the  limit- 


§  XII.]  AMSLER'S  PLANIMETER.  193 

ing  values  of  ^  differ  by  a  complete  revolution.  Hence  in  this 
case  equation  (2)  becomes 

A  —  Ttc?  ^^  bs  —  2  Tcbc, 

or  A  =  bs  +  7t{a^  —  2bc).'^ (4) 

In  another  form  of  the  planimeter  the  point  A  moves  in  a 
straight  line,  and  the  same  demonstration  shows  that  the  area 
is  always  equal  to  bs. 

Examples  XII. 

I.  The  involute  of  a  circle  whose  radius  is  a  is  drawn,  and  a  tangent 
is  drawn  at  the  opposite  end  of  the  diameter  which  passes  through  the 
cusp  ;  find  the  area  between  the  tangent  and  the  involute. 

a'n  (3  +  n"-) 


2.  Two  radii  vectoresof  a  closed  oval  are  drawn  from  a  fixed  point 
within,  one  of  which  is  parallel  to  the  tangent  at  the  extremity  of  the 
other  ;  if  the  parallelogram  be  completed,  the  area  of  the  locus  of  its 
vertex  is  double  the  area  of  the  given  oval. 

3.  Show  that  the  area  of  the  locus  of  the  middle  point  of  the  chord 
joining  the  extremities  of  the  radii  vectores  in  Ex.  2,  is  one  half  the 
area  of  the  given  oval. 


*  The  planimeter  is  usually  so  constructed  that  the  positive  direction  of  rotation 
is  with  the  hands  of  a  watch.  The  bar  b  is  adjustable,  but  the  distance  ^  C  is  fixed 
so  that  c  varies  with  b.  Denoting  AChy  q,  we  have  c  =^  q  —  \b,  and  the  constant 
to  be  added  becomes  C=-it  {a-  —  2bq  +  b-)  in  which  a  and  q  are  fixed  and  b  adjusta- 
ble.    In  some  instruments  q  is  negative. 

It  is  to  be  noticed  that  in  the  second  case  s  may  be  negative  ;  the  area  is  then 
the  numerical  difference  between  the  constant  and  bs. 


194  GEOMETRICAL  APTLICATIOXS.  [Ex.  XII. 

4.  Prove  that  the  difference  of  the  perimeters  of  two  parallel  ovals, 
whose  distance  is  b,  is  2nh,  and  that  the  difference  of  their  areas  is  the 
product  of  b  and  the  half  sum  of  their  perimeters. 

5.  A  lima'.on  is  formed  by  taking  a  fixed  distance  be  on  the  radius 
vector  from  a  point  on  the  circumference  of  a  circle  whose  radius  is  a  ; 
sho-.v  that  the  area  generated  by  b  when  b>  2a  \%  the  area  of  the  lima- 
9on  diminished  by  twice  the  area  of  the  circle,  and  thence  determine 
the  area  of  the  lima9on. 

7r(2<?'  -V  b^). 

6.  Verify  equation  (4),  Art.  167,  when  the  tracer  describes  the 
circle  whose  radius  '\%  a  ■\-  b. 

7.  Verify  the  value  of  the  constant  in  equation  (4),  Art.  167,  by 
determining  the  circle  which  may  be  described  by  the  tracer  without 
motion  of  the  wheel. 

8.  If,  in  the  motion  of  a  crank  and  connecting  rod  (the  line  of  motion 
of  the  piston  passing  through  the  centre  of  the  crank),  Amsler's  record- 
ing wheel  be  attached  to  the  connecting  rod  at  the  piston  end,  deter- 
mine .y  geometrically,  and  verify  by  means  of  the  area  described  by  the 
other  end  of  the  rod. 

9.  The  length  of  the  crank  in  Ex.  8  being  <?,  and  that  of  the  con- 
necting rod  b,  find  the  area  of  the  locus  of  a  point  on  the  connecting 
rod  at  a  distance  c  from  the  piston  end. 

nd'c 
~J~' 

10.  If  a  line  AB  of  fixed  length  move  in  a  plane,  returning  to  its 
original  position  without  making  a  complete  revolution,  denoting  the  areas 
of  the  curves  described  by  its  extremities  by  {A)  and  {B),  determine 
the  area  of  the  curve  described  by  a  point  cutting  AB  in  the  ratio 
m  :  «. 

7i{A)  +  m(B) 
m  +  n 


§  XII.] 


EXAMPLES. 


195 


II.  If  the  line  in  Ex.  10  return  to  its  original  position  after  making  a 
complete  revolution,  prove  HolditcJis  Theorem  ;  namely,  that  the  area  of 
the  curve  described  by  a  point  at  the  distance  c  and  c  from  A  and  B  is 


c\A)  +  c{B) 


c  ^  c' 


TtCC  . 


12.  Show  by  means  of  Ex.  11  that,  if  a  chord  of  fixed  length  move 
around  an  oval,  and  a  curve  be  described  by  a  point  at  the  distances 
c  and  c'  from  its  ends,  the  area  between  the  curves  will  be  ncc'. 


XIII. 

Approximate  Expr^essions  for  Areas  and  Volumes. 

(68.  When  the  equation  of  a  curve  is  unknown,  the  area 
between  the  curve,  the  axis  of  x,  and 
two  ordinates  may  be  approximately  ex- 
pressed in  terms  of  the  base  and  a  lim- 
ited number  of  ordinates,  which  are  sup- 
posed to  have  been  measured. 

Let  ABCDE  be  the  area  to  be  de- 
termined ;  denote  the  length  of  the  base 

by  2J1 ;  and  let  the  ordinates  at  the  ex- 

FiG.  23. 
tremities  and  middle  point  of  the  base 

be  measured  and  denoted  by  j'i,j'2,  and  jg.     Taking-  the  base  for 

the  axis  of  x,  and  the  middle  point  as  origin,  let  it  be  assumed 

that  the  curve  has  an  equation  of  the  form 

y  =  A  -^  Bx  +  Cx" -v  Dx^ ; (i) 


then  the  area  required  is 

,       f'        ,        ,        Ex'      Cx^      Dx^' 

A  —       ydx:^Ax^  —  +  —  + 


-h  -^3  4  ^-h 

in  which  which  A  and  C  are  unknown. 


=  -(6^+2a2),    .  (2) 


\g6 


GEOMETRICAL   APPLICATIONS. 


[Art.  i68. 


In   order  to  express   the    area  in  terms   of  the  measured 
ordinates,  we  have  from  equation  (i), 

)\^A^-  Bh  +  CJi-  +  Dh\ 

J'-i  =  -Ay 

j's=  A  -  Bh  +  Ch^-  Dh^\ 

whence  we  derive 

y\  ^  y^-2A  +  2Ch\ 

J'l  +  Ay%  +  J's  =  6A  +  2C1^ ; 


and  substituting  in  (2), 


^  =T-(j'i  +  4J'2  +  Js)- 

It  will  be  noticed  that  this  formula  gives  a  perfectly  ac- 
curate result  when  the  curve  is  really  a  parabolic  curve  of  the 
third  or  a  lower  degree. 

169.  If  the  base  be  divided  into  three  equal  intervals,  each 
denoted  by  h,  and  the  ordinates  at  the  extremities  and  at  the 
points  of  division  measured,  we  have,  by  assuming  the  same 
equation, 

A  =  ^\jdx=^J^{^A  +  ia?) (I) 

2 

From  the  equation  of  the  curve, 

iBh       gOl  _  270/^ 
2     "^      4  8      ' 

Bh      CJr      Dh^ 


Bh      Ch^      DW 


Fig.  24. 


,      iBh      gCh"^  ,  27DJt' 


§  XIII.]  SIMPSON'S  RULES.  1 9/ 

whence  j\  +  n  =  2A  +  - — ■ , 

2 

From  these  equations  we  obtain 

A  =  ~^'^  '^^^'^  "^  973 -J4 
16  ' 

and         ■  a^  =  71-J2-J3+J4  ^ 

4 

Substituting  in  equation  (i), 

A  =~-'(ji+  3J2+  3/3+^4). 


Simpson  s  Ricks. 

!70.  The  formulas  derived  in  Articles  168  and  169,  although 
they  were  first  given  by  Cotes  and  Newton,  are  usually  known 
as  Simpson  s  Rules,  the  following  extensions  of  the  formulas 
having  been  published  in  1743,  in  his  MatJicviatical  Disserta- 
tions. 

If  the  whole  base  be  divided  into  an  even  number  n  of 
parts,  each  equal  to  //,  and  the  ordinates  at  the  points  of  divis- 
ion be  numbered  in  order  from  end  to  end,  then  by  applying 
the  first  formula  to  the  areas  between  the  alternate  ordinates, 
we  have 

A  =-  ( j'l  +  \y-,  +  2j'3  +  4j'4  •  •  •  +  4J'«  +  yn  +  i). 

That  is  to  say,  the  area  is  equal  to  the  product  of  the  sum  of 
the  extreme  ordinates,  four  times  the  sum  of  the  even-num- 


198  GEOMETRICAL  APPLICATIONS.  [Art.  I70. 

bered  ordinates,  and  twice  the  sum  of  the  remaining  odd-num- 
bered ordinatcs,  multiplied  by  one  third  of  the  common  interval. 
Again,  if  the  base  be  divided  into  a  number  of  parts  divis- 
ible  by  three,  we  have,  by  applying  the  formula  derived  in 
Art.  169,  to  the  areas  between  the  ordinates  J1J4, 747,,  and  so  on, 

^  =  J  (j'l  +  3J'2  +  3J3  +  2/4  +  3J'5  •  •  •  +  3J'«  +  ;'«  +  .)• 

Cotes   Method  of  Approximation. 

171.  The  method  employed  in  Articles  168  and  169  is 
known  as  Cotes  Method.  It  consists  in  assuming  the  given 
curve  to  be  a  parabolic  curve  of  the  highest  order  which  can 
be  made  to  pass  through  the  extremities  of  a  series  of  equi- 
distant measured  ordinates. 

The  equation  of  the  parabolic  curve  of  the  «th  order  con- 
tains 71  -\-  \  unknown  constants;  hence,  in  order  to  eliminate 
these  constants  from  the  expression  for  an  area  defined  by  the 
curve,  it  is  in  general  necessary  to  have  n  -f-  i  equations  con- 
necting them  with  the  measured  ordinates.  Hence,  if  n  de- 
note the  number  of  intei'vals  between  measured  ordinates  over 
which  the  curve  extends,  the  curve  will  in  general  be  of  the 
«th  degree.* 


*  If  //  denotes  the  whole  base,  the  first  factor  is  always  equivalent  to  H 
divided  by  the  sum  of  the  coefficients  of  the  ordinates  ;  for  if  all  the  ordinates  are 
made  equal,  the  expression  must  reduce  to  Hy^.  Thus,  each  of  the  rules  for  an 
approximate  area,  including  those  derived  by  repeated  applications,  as  in  Art.  170, 
may  be  regarded  as  giving  an  expression  for  the  mean  ordinate.  The  coefficients 
of  the  ordinatcs,  according  to  Cotes'  method,  for  all  values  of  «  up  to  «  =  10,  may 
be  found  in  Bertrand's  Calcul  InL'i^ral,  pages  333  and  334.  For  example  (using 
detached  coefficients  for  brevity),  we  have,  when  n  —  4, 

^  =^  [7.  32,  12,  32,  7]; 
and  when  n  =  6, 

"^  ~  840  '■'*^'  ^^^'  ^^'  ^^^'  ^''  ^^^'  '^^^' 


§XIIL] 


THE   FIVE-EIGHT  RULE. 


199 


172.  For  example,  let  it  be  required  to  determine  the  area 
between  the  ordinates  y^  and  j'g,  in  terms  of  the  three  equi- 
distant ordinates  ji,  Jo  and  j'g,  the  common  interval  being  h. 

We  must  assume 

y  —  A  ^  Bx  +  Cx^\ 

then  taking  the  origin  at  the  foot  oi  y^. 


A  = 


*ydx=^  h 


.      Bh      Ch^- 
A  +■ —  +  — 
2  3  . 


from  which  A,  B  and  C  must  be   eliminated  by  means  of  the 
equations 

Ji  =  ^, 

ja  =  ^  +  ^/^  +  Ch^, 

ys  =  A  +  2B/1  +  4C/i\ 

Solving  these  equations,  we  obtain 
A  =yu 


Bh  = 
Ck 


Sn  +  47a  -  /3 


.2_Jl-2j'2+j3, 


If  we  make  a  slight  modification  in  the  ratios  of  these  last  coefficients  by  sub- 
stituting for  each  the  nearest  multiple  of  42,  we  have 

A  —  - —  [42,  210,  42,  252,  42,  210,  42], 
840 

(the  denominator  remaining  unchanged,  since  the  sum  of  the  coefficients  is  still 
840),  which  reduces  to 

^  =  ^[1-5,  1,6,  I,  5,  ij. 

This  result  is  known  as  Weddles"  Rule  for  six  intei-vals.  The  value  thus  given  to 
the  mean  ordinate  is  evidently  a  very  close  approximation  to  that  resulting  from 
Cotes'  method,  the  difference  being 

g^  [vi  +  >':  +  15  (73  +  ^'s)  -  6  ( ji  +  je)  -  2.oy,\ 


20O  GEOMETRICAL  APPLICATIOXS.  [Art.   1/2. 

and  substituting 

173.  It  is,  however,  to  be  noticed,  that  when  the  ordinates 
are  symmetrically  situated  with  respect  to  the  area,  if  n  is 
even,  the  parabolic  curve  may  be  assumed  of  the  {n  +  i)th 
degree.  For  example,  in  Art.  i68,  n  =  2,  but  the  curve  was 
assumed  of  the  third  degree.  Inasmuch  as  A,  B,  C  and  D 
cannot  all  be  expressed  in  terms  of  ji,  y-i,  and_)/3,  we  see  that  a 
variety  of  parabolic  curves  of  the  third  degree  can  be  passed 
through  the  extremities  of  the  measured  ordinates,  but  all  of 
these  curves  have  the  same  area.* 

Application   to  Solids. 

174.  \{  y  denotes  the  area  of  the  section  of  a  solid  perpen- 
dicular to  the  axis  of  -r,  the  volume  of  the  solid  is  lydx,  and 

*  This  circumstance  indicates  a  probable  advantage  in  making  «  an  even  num- 
ber when  repeated  applications  of  the  rules  are  made.  Thus,  in  the  case  of  six 
intervals,  we  can  make  three  applications  of  Simpson's  first  rule,  giving 

A  =  —[i,  4,  2,  4,  2,  4,  i], (I) 

or  two  of  Simpson's  second  rule,  giving 

^  =  ^  [i.  3,  3-  2,  3,  3,  i] (2) 

In  the  first  case,  we  assume  the  curve  to  consist  of  three  arcs  of  the  third  degree, 
meeting  at  the  extremities  of  the  ordinates  _j',j  and  jr.  ;  but,  since  each  of  these  arcs 
contains  an  undetermined  constant,  we  can  assume  them  to  have  common  tangents 
at  tl^  points  of  meeting.  We  have  therefore  a  smooth,  though  not  a  continuous 
curve.  In  the  second  case,  we  have  two  arcs  of  the  third  degree  containing  no 
arbitrary  constants,  and  therefore  making  an  angle  at  the  extremity  of  y^.  It  is 
probable,  therefore,  that  the  smooth  curve  of  the  first  case  will  in  most  cases  form  a 
better  approximation  than  the  broken  curve  of  the  second  case. 

In  confirmation  of  this  conclusion,  it  will  be  noticed  that  the  ratios  of  the 
coefficients  in  equation  (i)  are  nearer  to  those  of  Cotes'  coefficients  for«  =  6,  given 
in  the  preceding  foot-note,  than  are  those  in  equation  (2). 


§XIII.] 


APPLICATION    TO   SOLIDS. 


201 


therefore  the  approximate  rules  deduced  in  the  preceding  arti- 
cles apply  to  solids  as  well  as  to  areas.  Indeed,  they  may  be 
applied  to  the  approximate  computation  of  any  integral,  by 
putting  J  equal  to  the  coefficient  of  x  under  the  integral  sign. 

The  areas  of  the  sections  may  of  course  be  computed   by 
the  approximate  rules. 


Woolleys  Rule. 

175.  When  the  base  of  the  solid  is  rectangular,  and  the 
ordinates  of  the  sections  necessary  to  the  application  of  Simp- 
son's first  rule  are  measured,  we  may,  instead  of  applying  that 
rule,  introduce  the  ordinates  directly  into  the  expression  for 
the  area  in  the  following  manner. 

Taking  the  plane  of  the  base  for  the  plane  of  xy,  and  its 
centre  for  the  origin,  let  the  equation  of  the  upper  surface  be 
assumed  of  the  form 

z=A  ^Bx-^ Cy  +  D-x"  +  Exy  +  Ff  +  G.x^  +  Hx'^y  +  Ixf  +  Jf. 

Let  2h  and  2k  be  the  dimensions  of  the  base,  and  denote 
the  measured  values   of  r  as  indicated  in 
Fig.  25.     The  required  volume  is  '^ 


V  = 


a  dy  dx. 


This  double  integral  vanishes  for  every 
term  containing  an  odd  power  of  x  or  an 
odd  power  of  J  :  hence 

^,,      \DJi^k       ^FhB 
V=  AAhk  + + ; 


hk 


=  —\\2A  ^  aDJi^  ^- aFL^I (I) 

3 


202  GEOMETRICAL  APPLICA  TIOjVS.  [Art.  1 75. 

By  substituting  the  values  of  x  and  y  in  the  equation  of  the 
surface,  we  readily  obtain 

b2  =  A, (2) 

«i  +  ^3  +  ^1  +  ^3  =  4^^  +  A^^i^  +  AF^,     •     •     •    (3) 
^2  +  ^j  +  Ih  +  h  =  4A  +  2D}?  +  2FB.  ...    (4) 

From  these  equations  two  very  simple  expressions  for  the 
volume  may  be  derived ;  for,  employing  (2)  and  (4),  equation 
(i)  becomes 

2hk 
F=~-(^2  +  <5i  +  2^, +  <5'3+^2);     .     .     .     .  (4) 

and  employing  (2)  and  (3), 

hk 
F=  —  (^1  +  ^3  +  8^2  +  ^1  +  ^3) (5) 

Equation  (4)  is  known  as  WoolUys  Rule  ;  the  ordinates  employed 
are  those  at  the  middles  of  the  sides  and  at  the  centre  ;  in  (5), 
they  are  at  the  corners  and  at  the  centre. 


Examples  XIII. 

1.  Apply  Simpson's  Rule  to  the  sphere,  the  hemisphere,  and  the 
cone,  and  explain  why  the  results  are  perfectly  accurate. 

2.  Apply  Simpson's  Second  Rule  to  the  larger  segment  of  a  sphere 
made  by  a  plane  bisecting  at  right  angles  a  radius  of  the  sphere. 

~8~" 


§  XIIL]  EXAMPLES.  203 

3.  Find  by  Simpson's  Rule  the  volume  of  a  segment  of  a  sphere,  b 
and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 

?-(3'^=  +  3^'^+n 


4.  Find  by  Simpson's  Rule  the  volume  of  the  frustum  of  a  cone,  b 
and  c  being  the  radii  of  the  bases,  and  h  the  altitude. 

—  (b'^  +  be  +  c'). 
3 

5.  Compute  by  Simpson's   First  and  Second   Rules,  the  value  of 

,  the  common  interval  being  -^^  in  each  case. 

The  first  rule  gives  0.6931487,  and  the  second  rule  gives  0.6931505. 
The  correct  value  is  obviously  loge2  =  0.6931472. 

6.  Find  the  volume  considered  in  Art.  175,  directly  by  Simpson's 
Rule,  and  show  that  the  result  is  consistent  with  equations  (4)  and  (5). 

V=  —  [ai  +  as  +  Ci  +  C3  +  4  {a.2  +  bi  +b3  +  c^)  +  lOb^]. 

7.  Find,  by  elimination,  from  equations   (4)  and   (5),  Art.  175,  a 
formula  which  can  be  used  when  the  centre  ordinate  is  unknown. 

F=  —  [4{a,  +  b,  +  b,  +  c,)  -  (^1  +a,  +  ^1  +  c,)]. 
o 


204  MECHANICAL   APPLICATIONS.  [Art.    1 76. 


CHAPTER  IV. 
Mechanical  Applications. 


XIV. 
Definitions. 

176.  We  shall  give  in  this  chapter  a  few  of  the  applications 
of  the  Integral  Calculus  to  mechanical  questions. 

The  iuass  or  quantity  of  matter  contained  in  a  body  is  pro- 
portional to  its  weight.  When  the  masses  of  all  parts  of  equal 
v^olume  are  equal,  the  body  is  said  to  be  Jioviogcncoiis.  The 
factor  by  which  it  is  necessary  to  multiply  the  unit  of  volume 
to  produce  the  unit  of  mass  is  called  the  density,  and  usually 
denoted  by  y. 

In  the  following  articles  it  will  be  assumed,  when  not  other- 
wise stated,  that  the  body  is  homogeneous,  and  that  the  density 
is  equal  to  unity,  so  that  the  unit  of  mass  is  identical  with  the 
unit  of  volume.  When  the  mass  of  an  area  is  spoken  of,  it  is 
regarded  as  a  lamina  of  uniform  thickness  and  density,  and  the 
unit  of  mass  is  taken  to  correspond  with  the  unit  of  surface. 
In  like  manner  the  unit  of  mass  for  a  line  is  taken  as  identical 
with  the  unit  of  length. 

Statical  Moments. 

VII.  The  viovicnt  of  a  force,  with  reference  to  a  point,  is  the 
measure  of  the  effectiveness  of  the  force  in  producing  motion 
about  the  point.  It  is  shown  in  treatises  on  Mechanics,  that 
this  is  tlie  proihict  of  the  force  and  the  perpetidicular  from  the 
point  upon  the  line  of  application  of  the  force. 


§  XIV.]  STATICAL   MOMENTS.  205 

The  moment  of  the  sum  of  a  number  of  forces  about  a 
given  point  is  the  sum  of  the  moments  of  the  forces. 

The  statical  moDicnt  of  a  body  about  a  given  point  is  the 
moment  of  its  gravity  ;  the  force  of  gravity  being  supposed  to 
act  upon  every  part  of  the  body,  and  in  parallel  lines. 

178.  In  order  to  find  the  statical  moment  of  a  continuous 
body,  we  regard  the  body  as  generated  geometrically  in  some 
convenient  manner,  and  determine  the  corresponding  differen- 
tial of  the  moment. 

In  the  case  of  a  plane  area,  let  the  body  be  referred  to 
rectangular  axes,  and  let  gravity  be  supposed  to  act  in  the 
direction  of  the  axis  of  j'.  Then  the  abscissa  of  the  point  of 
application  is  the  arm  of  the  force  when  we  consider  the 
moment  about  the  origin.  Let  us  first  suppose  the  area  to  be 
generated  by  the  motion  of  the  ordinate/.  The  differential  of 
the  area  is  then  y  dx.     The  corresponding  element  of  the  sum, 

of  which  the  integral    y  dx  is  the  limiting  value,  see  Art.  99,  is 

JVA.r, (i) 

in  which  jjv  is  the  ordinate  corresponding  to  any  value  of  x 
intermediate  between  a  +  (r  — i)  A-r,  and  a  +  r  :\x.  It  is 
evident  that  the  arm  of  the  weight  of  the  element  (i)  is  such 
an  intermediate  value  of  x  ;  hence  the  moment  of  the  ele- 
ment is 

x^yr  t\x (2) 

The  whole  moment  is  therefore  the  limiting  value  of  a  sum^ 
of  the  form 

^  Xryr  A;ir. 


In  other  words,  it  is  the  integral 


xydx, (3) 


206 


MECHANICAL  APPLICA  TIONS. 


[Art.  178. 


in  which  the  differential  of  the  moment  is  the  product  of  the 
differential  of  the  area  and  the  arm  of  the  force,  which  in  this 
case  is  the  same  for  every  point  of  the  clement.  In  other 
words,  the  inovient  of  the  differential  is  the  differential  of  the 
vtoment. 

179.  As  an  illustration,  we  find  the  moment  of  a  semicircle 
(Fig.  26)  about  its  centre.  The  area  may  be 
generated  by  the  line  2r,  moving  from  ;tr  =  o  to 

Y 


Fig.  26. 


X  =  a.     The  equation  of  the  circle  being 

x"  ■\-  f  =  d\ 

the  differential  of  the  area  is 

2  4/(^2  _  ^^)  ^^_ 

The  moment  of  this  differential  is 
2  V{a^  —  x^)x  dx ; 


hence  the  whole  moment  is 


2f"  \'{c^  ~  x^)xdx  =  -  ?  {cC-  -^^fX  ='^^, 


Centres  of  Gravity. 

180.  If  a  force  equal  to  the  whole  weight  of  a  body  be 
applied  with  an  arm  properly  determined,  its  moment  may  be 
made  equivalent  to  the  whole  statical  moment  of  the  body. 
If  the  force  is  in  the  direction  of  the  axis  of;',  as  in  Fig.  26,  we 
have,  denoting  this  arm  by  "x, 

Ic  •  Area  =  Moment, 

_      Moment 
^  ~    Area     ' 


§  XIV.]  CEA'TEES   OF   GRAVITY.  20/ 

In  like  manner,  supposing  the  force  to  act  in  the  direction 
of  the  axis  of  x,  we  may  determine/  for  the  same  body. 

It  is  shown  in  treatises  on  Mechanics  that  the  point  deter- 
mined by  the  two  coordinates  x  and  y,  is  independent  of  the 
position  of  the  coordinate  axis.  This  point  is  called  the  centre 
of  gravity  of  the  area.  The  centre  of  gravity  of  a  volume  is 
defined  in  like  manner. 

181.  The  symmetry  of  the  form  of  a  body  may  determine 
one  oi  more  of  the  coordinates  of  its  centre  of  gravity.  Thus 
the  centre  of  gravity  of  a  circle  or  a  sphere  coincides  with  the 
geometrical  centre,  and  the  centre  of  gravity  of  a  solid  of  revolu- 
tion is  on  the  axis  of  revolution.  The  centre  of  gravity  of  the 
semicircle  in  Fig.  26,  is  on  the  axis  of  x ;  hence  to  determine 
its  position  we  have  only  to  find  x.  Dividing  the  moment 
of  the  semicircle  found  in  Art.  179  by  the  area  \ncB.,  we  have 

_      Aa 

X  r=z  —  . 

(82.  In  finding  the  moment  of  the  semicircle  (Art.  179),  we 
regarded  the  area  as  generated  by  the  double  ordinate  2y,  and 
the  differential  of  the  moment  was  found  by  multiplying  the 
differential  of  the  area  by  x,  which  is  the  arm  of  the  force  for 
every  point  of  the  generating  line. 

We  may,  however,  derive  the  moment  from  the  differential 
of  area, 

^'^J, (0 

since  the  area  may  be  generated  by  the  motion  of  the  abscissa 
X  from  J  =  —  a  to  y  =  a.  But  in  this  case  to  find  the  moment 
of  the  differential  we  must  multiply  it  by  the  distance  of  its 
centre  of  gravity  from  the  given  axis.  The  centre  of  gravity  of 
the  line  x  is  evidently  its  middle  point,  hence  the  required  ^rm 
is  ^x.     Therefore  the  differential  of  the  moment  is 

x^  dy  ,  . 

-z^) (2) 


20S  MECHANICAL   APPLICATIONS.  [Art.   1 82. 

and  consequently  the  whole  moment  is 

This  result  is  identical  with  that  derived  in  Art.  179. 


Polaj'  Formulas, 

183.  When  polar  formulas  are  employed,  r  and  B  being 
coordinates  of  the  curved  boundary  of  the  area,  the  element  is 
\r^  dS.  Since  this  element  is  ultimately  a  triangle,  we  employ 
the  well  known  property  of  triangles  ;  that  the  centre  of  gravity 
is  on  a  medial  line  at  two-thirds  the  distance  from  the  vertex 
to  the  base. 

The  coordinates  of  the  centre  of  gravity  of  the  element  are, 
therefore, 

2       .  2 

-rsin6'  and  —rcosO. 


3 


Hence  we  have  the  formula 


Urcosd^r^dO      2    [;^cos^^^ 


.*•  = 


Ur'dt) 


^1' 

fr^sin^rt'^ 


r^df^ 


3 


and  similarly  J'  =     ■ 

3       jr^  de 


§  XIV.]  POLAR   FORMULAS.  209 

184.  To  illustrate,  let  us  find  the  centre  of  gravity  of  the 
area  enclosed  by  the  lemniscata 

^2  —  ^j,2  ^Qg  26, 


(cos  2^) 'cos  Odd 

Whence     .r  = =  — 

3  -  3 

f'cos2^./(y 


(cos  2O)  COS  Odd. 


Put  cos  26  —  cos"  4>,         whence  sin  (^  =  ^2  sin  d, 

and  \'2  cos  S  dQ  —  cos  ^  ^^, 


-       2i'2       f2         ,  ,    ,,        2\'2     l-l     TT  t  2 

3       Jo  "^         3       4-2    2  b 


Solids  of  Revolution. 

'  185.  To  find  the  centre  of  gravity  of  a  solid  of  revolution, 
we  take  the  axis  of  revolution  as  the  axis  of  x,  and  the  circle 
whose  area  is  irf  as  the  generating  element.  Replacing  y  in 
equation  (3),  Art.  178,  by  this  expression,  we  have  for  the  stati- 
cal moment 

71     xjr  dx, 
and  for  the  abscissa  of  the  centre  of  gravity 

,rj'^  dx 
X  —  L"-, . 

y  dx 


£• 


210 


MECHANICAL   A  P PLICA  TIONS. 


[Art.  186. 


186.  To  illustrate,  wc  find  the  centre  of  gravity  of  a  spheri- 
cal segment  whose  height  is  li.  In  this  case,  taking  the  origin 
at  the  vertex  of  the  segment,  and  denoting  the  radius  of  the 
sphere  by  a,  we  have 


^1^ 


y 


■2  ^.  2ax  -  ^. 


Hence 


{2ax'  —  x^)  dx      -ax^  -  -x^ 
{2a  X  —  x^)dx       ax 


-X 

3    J 


//   Sa  -  ih 

4.     Vi  —  h  ' 


If  the  centre  of  gravity  of  the  surface  of  the  scgvient  be  re- 
quired, bince  the  differential  of  the  surface  is  27Ty  ds,  we  easily 
obtain  the  general  formula 


x  = 


xy  ds 

'  o 

-yjt        ' 
lyds 


and,  in  this  case  the  curve  being  a  circle,/  ds  =  a  dx ;  hence, 
substituting,  we  have 

X  =  \h. 


The  Properties  of  Pappus. 

187.  Let  a  solid  be  generated  by  the  revolution  of  any  plane 
figure  about  an  exterior  axis  in  its  own  plane.  It  is  required 
to  determine  the  volume  and  the  surface  thus  generated. 

It  is  evident  that  this  solid  may  also  be  generated  by  a 
variable  circular  ring  w^hose  centre  moves  along  the  axis  of 
revolution  ;  denoting  by  y^  and  y^  corresponding  ordinates  of 


XIV.]  THE  PROPERTIES  OF  PAPPUS.  211 


the  outer  and  inner  circles  respectively,  the  area  of  this  ring  is 
'^{yx  —  yi).     Hence 

V^  n\{y^--yi)  dx  =  2;rpA±Zi(_j,^  -j,;)dx. 

But   this  integral  is  the  statical  moment  of  the  given  figure, 

since  j'l  —  j'^  is  the  generating  element  of  its  area,  and  — — ^is 

the  corresponding  arm.     Denoting  the  area  of  the  figure  by  A, 
we  may  therefore  write 

V=27ryA; 

that  is,  //le  volume  is  tJie  product  of  the  area  of  the  figure  and  the 
path  described  by  its  ccfitre  of  gravity. 

The  surface  {S)  of  this  solid  is,  by  Art.  149, 


S  =  2n\yds  =.27T 


dSf 


ii  J  denotes  the  ordinate  of  the  centre  of  gravity  of  the  arc  s. 


Hence  we  have  5  =  2ny-arc ; 

that  is,  tJie  surface  is  the  product  of  the  lengtli  of  the  arc  into 
the  path  described  by  the  centre  of  gravity. 

These  theorems  are  frequently  called  the  properties  of  Gul- 
dinus ;  they  are,  however,  due  to  Pappus,  who  published  them 
1588. 

It  is  obvious  that  both  theorems  are  true  for  any  part  of 
a  revolution  of  the  generating  figure. 


212  MECHAXICAL   APPLICATIONS.  [Ex.  XIV, 


Examples    XIV. 

1.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
parabola  y  =  ^mx  and  the  double  ordinate  corresponding  to  the 
abscissa  a. 

5 

2.  Find  the  centre  of  gravity  of  the  area  between  the  semi-cubical 
parabola  ay^  =  x^  and  the  double  ordinate  which  corresponds  to  the 
abscissa  a. 

'   -  _  5^ 


3.  Find  the  ordinate  of  the  centre  of  gravity  of  the  area  between 
the  axis  of  ~v  and  the  sinusoid  j  =  sin  x,  the  limits  being  .r  =  o  and 

4.  Find  the  coordinates  of  the  centre  of  gravity  of  the  area  be- 
tween the  axes  and  the  parabola 


/\i_ 


,7;    +  ^1)    =  ■• 

—      a  ■,  -      b 

A-  =  — ,  and  v  =  -  . 
5  '        5 

5.  Find  the    centre   of  gravity  of   the  arja  between    the   cissoid 
y^  (a  —  x)  =  x^  and  its  asymptote. 
Solution  : — 

Denoting  the  statical  moment  by  M  and  the  area  by  A^ 

-,     f"  A^ dx  I. ,        ,,n«       r«  3. .         w  , 

M  =      =  —  2JCS  ia  —  xy-      +  1^      •v='  {a  —  xP  dx 


f"   x^ dx              I.,         ,,n 

a 

+ 

0 

5 

x-i  {a  —  x) 

:  —      2A-»  ya  —  .vj- 

=  5'^  •  ^^  -  5-1^ ; 

\M  —  ^  A,                     hence 
0 

-=?• 

213 


§  XIV.]  EXAMPLES. 

6.  Find  the  centre  of  gravity  of  the  area  between  the  parabola 
v^  =  4ax  and  the  straight  line  7  =  mx. 

—        8a  ,  -      2« 

-v  =  — —    and  J'  =  — . 
Sm  m 

7.  Find  the  centre  of  gravity  of  the  segment  of  an  ellipse  cut  off 
by  a  quadrantal  chord. 

—       2a  .    -      2         b 

-V  =  -  • and    y  =  -  • . 

I     7t  —  2  ■'       I     7t—  2 

8.  Given  the  cycloid, 

y  —  a{i  —  cos  '/•)>  X  =  a  {ip  —  sin  rp)  , 

find  the  distance  of  its  centre  of  gravity  from  the  base. 

>  =  f- 

9.  Find  the  centre  of  gravity  of  the  area  enclosed  between  the 
positive  directions  of  the  coordinate  axes  and  the  four-cusped  hypo- 
cycloid 


.r^  +  J^'^^  =  Or' 
Put  X  =  a  cos'  b,  and y  =  a  sin^  0. 


2^(ia 

x=y  =  -^ . 

315^ 


10.  Find  the  centre  of  gravity  of  the  area  enclosed  by  the  cardioid 
r  =  <z  (l  —  cos  0). 


II.  Find  the  centre  of  gravity  of  the  sector  of  a  circle  whose  radius 
is  a,  the  angle  of  the  sector  being  2a. 

-       2  a?>\y\oc 
Use  the  method  of  Art.  183.  ^  — • 


3 


a 


214  MECHANICAL  APPLICATIOXS.  [Ex.  XIV. 


12.  Find  the  centre  of  gravity  of  the  segment  of  a  circle,  the  angle 
subtended  being  2«  and  the  radius  of  the  circle  a. 


Solution : — 


2 
X  =  - 


J  ^  cola  _  2«  Sin  a  _  Chord 

Area  3  Area    ~  12  Area" 


13.  Find  the  centre  of  gravity  of  a  circular  ring,  the  radii  being  a 
and  <7„  and  the  angle  subtended  2  a. 

-  _  2    a^  —  a^     sin  ce 

~  3    (f  —  a^        a 

14.  Find  the  centre  of  gravity  of  a  circular  arc,  whose  length  is  2s. 

Solution : — 

We  have  in  this  case,  taking  the  origin  at  the  centre  and  the  axis 
of  X  bisecting  the  arc. 


-    f^ 

X  —  •'  -' 

~  I: 


ds 
Is 


Put  X  —  a  cos  0,          then  ds  =  a  dO,  and  denoting  by  a  the 

angle  subtended  by  s,  we  have 


a  I     cos  0  d^j 

asm  a        c 


2c  being  the  chord. 


§  XIV.]  EXAMPLES,  215 

15.  Find  the  coordinates  of  the  centre  of  gravity  of  arc  of  the  semi- 
cycloid  whose  equations,  referred  to  the  vertex,  are 

X  =■  a{\  —  cos  ip),  and  y  =^  a  (ip  +  sin  tp). 

X  =  — ,  and  y  =  (tt  —  -  j  a. 

16.  Find  the  centre  of  gravity  of  the  arc  between  two  successive 
cusps  of  the  four-cusped  hypocycloid 

2.  i  2 

_  -_  2« 

^~      "J' 

17.  Find  the  position  of  the  centre  of  gravity  of  the  arc  of  the  semi- 
cardioid 

r  =  a  {i  —  cos  ^). 

-  4a        .  -      Aa 

X  = ,  and  y  =  —  . 

5  5 

18.  A  semi-ellipsoid  is  formed  by  the  revolution  of  a  semi-ellipse 
about  its  major  axis  ;  find  the  distance  of  the  centre  of  gravity  of  the 
solid  from  the  centre  of  the  ellipse. 

X  =  ^  . 


19.  Find  the  centre  of  gravity  of  a  frustum  of  a  paraboloid  of 
revolution  having  a  single  base,  A  denoting  the  height  of  the  frustum. 

-  2/i 


20.  A  paraboloid  and  a  cone  have  a  common  base  and  vertices  at 
the  same  point  ;  find  the  centre  of  gravity  of  the  solid  enclosed 
between  them. 

The  centre  of  gravity  is  the  middle  point  of  the  axis. 


2l6  MECHANICAL  APPLICATIONS.  [Ex.  XIV. 

21.  Find  the  centre  of  gravity  of  a  hyperboloid  whose  height  is  K 
the  generating  curve  being 

y  =  m  {2 ax  +  -x"). 

-  _  //    8a  +  zh 

~  4     3^  +  '^  " 

22.  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revolution 
of  the  sector  of  a  circle  about  one  of  its  extreme  radii. 

The  height  of  the  cone  being  denoted  by  //,  and  the  radius  of  the 
circle  by  a,  we  have 

^  =  |(a+/0. 

23.  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revolution 
about  the  axis  of  x  of  the  curve 

cry  =  ax^  —  a-', 
between  the  limits  o  and  a. 

8  • 

24.  A  solid  is  formed  by  revolving  about  its  axis  the  cardioid 

r  =  a  {i  —  cosO)  ; 

find  the  distance  of  the  cusp  from  the  centre  of  gravity. 

—       16a 

'^~   15  ■ 

25.  Determine  the  position  of  the  centre  of  gravity  of  the  volume 
included  between  the  surfaces  generated  by  revolving  about  the  axis 
of  X  the  two  parabolas 

y  =  mx,  and  y'^  =  m'  {a  —  x). 

-  a    m  +  2m 
X  = 


3     m  +  m 


§  XIV.]  EXAMPLES.  217 

26.  Find  the  centre  of  gravity  of  a  rifle  bullet  consisting  of  a  cylin- 
der two  calibers  in  length,  and  a  paraboloid  one  and  a  half  calibers  in 
length  having  a  common  base,  the  opposite  end  of  the  cylinder  con- 
taining a  conical  cavity  one  caliber  in  depth  with  a  base  equal  in  size 
to  that  of  the  cylinder. 

The  distance  of  the  centre  of  gravity  from  the  base  of 
the  bullet  is  \%\  calibers. 

27.  A  solid  formed  by  the  revolution  of  a  circular  segment  about 
its  chord  is  cut  in  halves  by  a  plane  perpendicular  to  the  chord  ; 
determine  the  centre  of  gravity  of  one  of  the  halves.  This  solid  is 
called  an  ogival. 

Denoting  by  2a  the  angle  subtended  by  the  chord,  and  by  a  the 
radius  of  the  circle,  the  distance  of  the  centre  of  gravity  from  the 
base  is 

—  _  d     44  sin-  a  4-  sin^  2(t  4-  32  (cos  201  —  cos  a) 
16  sin  a  (2  -f  cos"  a)  —  yx  cos  a 

28.  Find  the  centre  of  gravity  of  the  surface  of  the  paraboloid 
formed  by  the  revolution  about  the  axis  of  x  of  the  parabola 

a  denoting  the  height  of  the  paraboloid. 

-.  _  I     (3^  —  2111)  (a  4-  my-i  4-  2/«8 
5  {a  -h  my  —  m-i 

29.  Find  the  centre  of  gravity  of  the  surface  generated  by  the  revo- 
lution of  a  semi-cycloid  about  its  axis,  the  equations  of  the  curve 
being 

.V  =  a  (i  —  cos  ^),  and  _y  =  a  (^'  4-  sin  ^•). 

2a     Y^TT  —  % 

X  =  —  •  -^ 

15      3^-  4  ' 


2l8  MECHANICAL  APPLICATIONS.  [Ex.  XIV. 

30.  Find  the  centre  of  gravity  of  the  surface  generated  by  the  revo- 
lution about  its  axis  of  one  of  the  loops  of  the  lemniscata 

r'  zn  d'  cos  2O. 

—         2+1^2 

X  = ; a. 


31.  A  cardioid  revolves  about  its  axis  ;  find  the  centre  of  gravity 
of  the  surface  generated,  the  equation  of  the  cardioid  being 


r  =  a  {1  —  cosO). 


x-=  ^ —  , 
6; 


32.  A  ring  is  generated  by  the  revolution  of  a  circle  about  an  axis 
in  its  own  plane  ;  c  being  the  distance  of  the  centre  of  the  circle 
from  the  axis,  and  a  the  radius,  determine  the  volume  and  surface 
generated. 

F=  21V cd\  and  S—  ^Tt-ca. 

^^.  A  triangle  revolves  about  an  axis  in  its  plane  ;  a^,  a^,  and  a^, 
denoting  the  distances  of  its  vertices  from  the  axis,  determine  the  vol- 
ume generated. 

27rA  ,  . 

V  = (^1  +  a,  +  a^). 

3 

34.  Find  the  volume  of  a  frustum  of  a  cone,  the  radii  of  the  bases 
being  Ui  and  a-j,  and  the  height  h. 

•3 

35.  Find  the  volume  and  surface  generated  by  the  revolution  of  a 
cycloid  about  its  base. 

y=  ^Tta,  and  o  = . 


XV.]  MOMENTS  OF  INERTIA.  2ig 


XV. 

Afomen/s  of  Inertia. 

188.    When  a  body  rotates  about  a  fixed  axis,  the  velocity 
of  a  particle  at  a  distance  r  from  the  axis  is 

dco 

in  which  go  is  the  angle  of  rotation.  The  force  which  acting 
for  a  unit  of  time  would  produce  this  motion  in  a  mass  jh  is 
measured  by  the  momentum 

doo 

mr  —— . 
dt 

The  moment  of  this  force  about  the  axis  is  therefore 

o  doo 
dt 

The  sum  of  these  moments  for  all  the  parts  of  a  rigid  system  is 

doj  „         , 
—r  -^  A  mr' , 
dt  ' 

since  the  angular  velocity, —p,  is  constant.      In  the  case  of  a 

dt 

continuous  body  this  expression  becomes 

dco  f  ,  , 
-—  xr  dm. 
dt  J 

in  which  dm  is  the  differential  of  the  mass.     The  factor 

r^  djn, 


220  MECHANICAL  APPLICATIONS.  [Art.   1 88. 

which  depends  upon  the  shape  of  the  body,  is  called  its  mo- 
inent  of  inertia,  and  is  denoted  by  /. 

189.  When  the  body  is  homogeneous,  dm  is  to  be  taken 
equal  to  the  differential  of  the  line,  area,  or  volume,  as  the  case 
may  be.  For  example,  in  finding  the  moment  of  inertia  of  a 
straight  line  whose  length  is  2a,  about  an  axis  bisecting  it  at 
right  angles,  we  let  x  denote  the  distance  of  any  point  from 
the  axis;  then  dm  =  dx,  hence  we  have 


I=['  ,^dx  =  '-^=^-''l 

)-a  ^  12 


Again,  in  finding  the  moment  of  inertia  of  the  semi-circle  in 
figure  25,  about  the  axis  oi y,  let  d}n=  2ydx;  then,  since  every 
point  of  the  generating  line  is  at  the  distance  x  from  the  axis, 
the  moment  of  inertia  is 

7=2     yx^  dx  =  2       ^/{c?  —  x^)  x^  dx . 

Jo  Jo 

Putting  X  =  a  sin  B,  we  have 

1 
/  =  2a^  \  cos2  B  sin2  d  dS  =  ~  . 

Jo  «J 

T/ie  /Radius  of  Gyration. 

190.  If  the  whole  mass  of  the  body  were  situated  at  the 
distance  k  from  the  axis,  its  moment  of  inertia  would  be  Ic^m. 
Now,  if  k  is  so  determined  that  tJiis  vwmcnt  shall  be  equal  to 
the  actual  moment  of  inertia  of  the  body,  the  value  of  k  is  tJie 
radius  of  gyratiofi  of  the  body  with  reference  to  the  given 
axis.     Hence 

^  _  Moment  of  inertia 
Mass  * 


§  XV.]  THE  RADIUS  OF  GYRATION.  221 

Thus,  for  the  radius  of  gyration  of  the  line  2a,  whose  moment 
of  inertia  is  found  in  the  preceding  article,  we  have 

/^  =-  ,  or  k=  —  \ 

3  V3 

and  for  the  radius  of  gyration  of  the  semi-circle,  whose  area 

is  \7lC^, 

j^      a^  y      a 

k-  —  ~  ,  or  k=  ~. 

4  •  2 

It  is  evident  that  this  expression  is  also  the  radius  of  gyra- 
tion of  the  whole  circle  about  a  diameter,  for  the  moment  of 
inertia  of  the  circle  is  evidently  double  that  of  the  semi-circle, 
and  its  area  is  also  double  that  of  the  semi-circle. 

191.  It  is  sometimes  convenient  to  use  modes  of  generating 
the  area  or  volume,  other  than  those  involving  rectangular 
coordinates.  For  example,  let  it  be  required  to  find  the  radius 
of  gyration  of  a  circle  whose  radius  is  a,  about  an  axis  passing 
through  its  centre  and  perpendicular  to  its  plane.  This  circle 
may  be  generated  by  the  circumference  of  a  variable  circle 
whose  radius  is  r,  while  r  passes  from  o  to  a.  The  differential 
of  the  area  is  then  27ir  dr,  and  the  moment  is 

I  =  27t\    f^  dr  =  —  . 
Jo  2 

Dividing  by  the  area  of  the  circle,  we  have 

a^ 

2 

192.  Again,  to  find  the  radius  of  gyration  of  a  sphere 
whose  radius  is  a  about  a  diameter.  In  order  that  all  points 
of  the  elements  shall  be  at  the  same   distance  from  the  axis 


222  MECHANICAL  APPLICATIONS.  [Art.  I92. 

we  regard  the  sphere  as  generated  by  the  surface  of  a  cyHnder 
whose  radius  is  x,  and  whose  altitude  is  2y.  The  surface  of 
this  cylinder  is  therefore  4^,17.  The  differential  of  the  volume 
is  /^Ttxy  dx,  and  the  moment  of  inertia  is 


x^y  dx  =  /^7i\  V{a-  —  x)  x^dx. 


Putting X  =  asin  6, 

IT 

I  =  47TCV- \\m^  d  cos^  e  dd 

J  o 


Dividing  by  - —  ,  the  volume  of  the  sphere,  we  have 


5 


Radii  of  Gyration  about  Parallel  Axes. 

193.  The  moment  of  inertia  of  a  body  about  any  axis  exceeds 
its  moment  of  inertia  about  a  parallel  axis  passing  tJirougJi  the 
centre  of  gravity ^  by  the  product  of  the  mass  and  the  square  of 
the  distance  between  the  axes^ 

Let  h  be  the  distance  between  the  axes.  Pass  a  plane 
through  the  element  dm  perpendicular  to  the  axes,  and  let  r 
and  ri  be  the  distances  of  the  element  from  the  axes.  Then, 
r,  rj,  and  h  form  a  triangle ;  let  6  be  the  angle  at  the  axis 
passing  through  the  centre  of  gravity,  then 

r^  =  r'^  +  U^  —  2rji  cos  B (i) 


§  XV.]       RADII  OF  GYRATION  ABOUT  PARALLEL  AXES.        223 

The  moment  of  inertia  is  therefore 

I  r^ihn  —     rl  dm  +  H^'m  —  2h     7\  cos  (^ dm  .     .     .     (2 ) 

Now  ri  and  6  are  the  polar  coordinates  of  dm,  in  the  plane 
which  is  passed  through  the  element;  hence  the  last  integral  in 
equation  (2)  is  equivalent  to 

—  2h     X  dm. 


But     X  dm  is  the  statical  moment  of  the  body  about  the  axis 

passing  through  the  centre  of  gravity.  Nov.'  from  the  defini- 
tion of  the  centre  of  gravity,  this  moment  is  zero  ;  hence, 
equation  (2)  reduces  to 

'T  dm  =    r^  dm  +  Jrm  .     .     .     .     .     .  (3 

Introducing  the  radii  of  gyration,  we  have  also 

}^^k~-^h' (4) 

I94-.  As  an  application  of  this  result,  we  shall  now  find  the 
moment  of  inertia  of  a  cone  whose  height  is  //,  and  the  radius 
of  whose  base  is  a^  about  an  axis  passing  through  its  vertex 
perpendicular  to  its  geometrical  axis.  Taking  the  origin  at 
the  vertex  of  the  cone,  the  axis  of  x  coincident  with  the  geo- 
metrical axis,  and  a  circle  perpendicular  to  this  axis  as  the 
generating  element,  we  have  for  the  area  of  this  element  ;r/^ 
and    for  its   radius  of  gyration   about   a   diameter  parallel  to 

the  given  axis, '—  . 


224  MECHANICAL  APPLICATIONS.  [Art.  I94- 

The  distance  between  these  axes  being  x,  the  proposition 
proved  in  the  preceding  article  gives  an  expression  for  the 
radius  of  gyration  of  the  element  about  the  given  axis;  viz., 

x^  +—  .     Replacing  r^,   in  the  general  expression  for  /  (Art. 

4 
188),  by  this  expression,  and  substituting  iox  dm  the  differen- 
tial Tty^  dx,  we  have 


{f^^-^fdx, 


(IX 

in  which  j'  =  ^-  .         Therefore 
n 


and  since  V  — , 

3 

To  find  the  square  of  the  radius  of  gyration  about  a 
parallel  axis  through  the  centre  of  gravity,  we  have 

To  find  the  moment  of  inertia  of  a  right  cone  about  its 
geometrical  axis  wc  employ  the  same  generating  element  as 
before  ;  but  in  this  case  the  square  of  the  radius  of  gyration  is 

—  .      Hence 
2 

7r<7-*  V' 


2 

4 


^=-l\^''-^-T,A/'"' 


§  XV.]      RADII  OF  GYRATION  ABOUT  PARALLEL  AXES.         22$ 


therefore 


/= ,     whence     J^  =  ^, 

10  lO 


Polai'  Moments  of  Inertia. 

195.  In  the  case  of  a  plane  area,  when  the  axis  of  rotation 
passes  through  the  origin,  we  have 

r^  =  ;i-2  -[-  y^,  hence    r^  dm  ~    {x^  +  y-)  dtii, 


therefore  /=  \.-^  dm  + 


=    ,r^  dm  +     j'^  dm 


that  is,  the  sum  of  the  moments  of  inertia  of  a  plane  area  about 
tzuo  axes  in  its  ozun  plane  at  right  angles  to  each  other  is  eqiial  to 
the  moment  of  inertia  about  an  axis  through  the  origin  perpendicu- 
lar to  the  plane.  I  in  the  above  equation  is  called  the  polar 
moment  of  inertia. 

In  the  case  of  the  circle,  since  the  moment  is  the  same 
about  every  diameter,  the  polar  moment  is  twice  the  moment 
about  a  diameter ;  that  is,  denoting  the  former  by  Ip  and  the 
latter  by  /,„  we  have 


See  Art.  191. 


T  T  ^^ 


Examples  XV. 

I.  Find  the  radius  of  gyration  of  a  circular  arc  {2s)  about  a  radius 
passing  through  its  vertex. 


226  MECHANICAL  APPLICATIONS.  [Ex.  XV. 

Solution  : — 

Taking  the  origin  at  the  centre,  and  the  axis  of  x  bisecting  the  arc, 
and  denoting  by  2ix  the  angle  subtended  by  2s,  we  have 


mk '  =  ['  /  (fs  =  a'  [*  sin'  0  do. 


,„       a  f         sin  20! 
m  =  2aa  .'.  ^  =  -  I  i  — 

2    \  2a 


2.  Find  the  radius  of  gyration  of  the  same  arc  about  the  axis  of  y, 
and  thence  about  a  perpendicular  axis  through  the  centre  of  the 
circle.  ^  =  ^• 

3.  Find  the  radius  of  gyration  of  the  same  arc  about  an  axis  through 
its  vertex  perpendicular  to  the  plane  of  the  circle. 

See  Ex.  XIV.,  14,  and  denote  by  c  the  subtending  chord. 

k''  =  2a'(i-~ 

4.  Find  the  moment  of  inertia  of  the  chord  of  a  circular  arc,  in 
terms  of  the  diameter  parallel  to  it,  and  its  angular  distance  from  this 
diameter. 

See  Arts.  189  and  193.  /=—  (3  cos  a-  —  cos3n'). 

5.  Find  the  radius  of  gyration  of  an  ellipse  about  an  axis  through 
its  centre  perpendicular  to  its  plane. 

Find  the  radius  of  gyration  about  the  major  axis  and  about  the  minor 
axis,  and  apply  Art.  195. 

6.  Find  the  radius  of  gyration  of  an  isosceles  triangle  about  a  per- 
pendicular let  fall  from  its  vertex  upon  the  base  {2b), 

k'  =  '- 

6' 


§  XV.]  EXAMPLES.  227 

7.  Find  the  radius  of  gyration  about  the  axis  of  the  curve,  of  the 
area  enclosed  by  the  two  loops  of  the  lemniscata 


r   =  (2'  cos  29. 


^'=^(3^-8). 


8.  Find  the  radius  of  gyration  of  a  right  triangle,  whose  sides  are  a 
and  b.  about  an  axis  through  its  centre  of  gravity  perpendicular  to  its 
plant' 


9.  Find  the  radius  of  gyration  of  a  portion  of  a  parabola  bounded . 
by  a  double  ordinate  perpendicular  to  the  axis,  about  a  perpendicular 
to  its  plane  passing  through  its  vertex. 

10.  Find  the  radius  of  gyration  of  a  cylinder  about  a  perpendicular 
that  bisects  its  geometrical  axis,  2/  being  the  length  of  the  cylinder, 
and  a  the  radius  of  its  base. 

,0       ci'       I' 
4       3 

11.  Find  the  radius  of  gyration  of  a  concentric  spherical  shell  about 
a  tangent  to  the  external  sphere,  the  radii  being  a  and  b. 

7^;"  —  5«V  —2b'' 


k'  = 


S{a^-n 


12.  Find  the  radius  of  gyration  of  a  paraboloid  of  revolution  about 
its  axis,  in  terms  of  the  radius  {b)  of  the  base. 

,.      b' 


13.  Find  the  moment  of  inertia  of  an  ellipsoid  about  one  of   its 
principal  axes. 

15 


228  MECHANICAL  APPLICATIONS.  [Ex.  XV, 

14.  Find  the  radius  of  gyration  of  a  symmetrical  double  convex  lens 
about  its  axis,  a  being  the  radius  of  the  circular  intersection  of  tne 
two  surfaces,  and  b  the  semi-axis. 

,  _  //  +  ^d'b^  +  \oa* 


\o{b'  +  la-) 


15.  Find  the  radius  of  gyration  of  the  same  lens  about  a  diameter 
to  the  circle  in  which  the  spherical  surfaces  intersect. 

^  _  \oa'  +  x^d'b^  4-  it* 


THE   END. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 

This  book  is  DUE  on  the  last  date  stamped  below. 

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JUL   'L 
OCT   8       «4J 

NOV    ^  i  1947 
jUN  1  5  195C 

|iB>ttiill^ 

ecri4  w^ 

^^'V 13/95, 
iOV  1  9  1^5! 

APR  2 1  1952 

|\U6  2  8  1W3 

JUN    4       1954 

Form  L9-25m-8,'46(9852)444 


JAN 

JUN 


t    9 

1  n 


JUL  2  7  1955 
SEP  1  4  195tf 


^m 


wMm 


'  ii  n 

;■■:' iV  a 

:■■■■        ,  » 


